Phase retrieval using through-focus images in Lorentz transmission electron microscopy

Phase retrieval using through-focus images in Lorentz transmission electron microscopy Abstract In this study, we report on phase retrieval by the maximum likelihood method with conjugate gradient method (CG-MAL) using through-focus images in Lorentz transmission electron microscopy (LTEM). The method was evaluated using 32 simulated and experimentally obtained through-focus images of magnetic bubbles; these images were collected under a defocus range from approximately −3 mm to 3 mm. Consequently, we obtained the magnetic domain structures of the magnetic bubbles in both the simulation and the LTEM experiment. Furthermore, the CG-MAL method showed better convergence behavior than other iterative phase retrieval methods. Therefore, the method can also be widely and effectively applied to the observation of magnetic domain structures other than magnetic bubbles when highly defocused through-focus images are used. Lorentz transmission electron microscopy, phase retrieval, focal-series reconstruction, iterative wave function reconstruction, maximum likelihood method, magnetic domain structure Introduction In Lorentz transmission electron microscopy (LTEM) [1], phase retrieval using through-focus images [2] is a powerful technique for observing magnetic domain structures within samples in real space. Although this approach has the advantage of enabling phase retrieval without any additional devices, it also has several disadvantages compared with other phase retrieval techniques. For instance, the time resolution of the focal-series reconstruction methods is usually lower than that of techniques such as differential phase contrast imaging in scanning TEM (DPC-STEM) [3, 4] because the focal-series reconstruction methods usually require multiple defocused images and image processing. More importantly, the quantitative performance of the focal-series reconstruction methods is often inferior to that of off-axis electron holography [5, 6]. Performing the focal-series reconstruction more quantitatively requires utilizing highly defocused images in a wide defocus range because the magnitude of the observed image contrast generally depends on the magnitude of the defocus value. A phase retrieval method that utilizes a transport-of-intensity equation (TIE) [7] (i.e. a TIE method) is the most popular focal-series reconstruction method in LTEM; it generally involves an in-focus image and two defocused images [8]. However, the TIE method has several problems attributed to the fact that the method can appropriately deal with only three LTEM images obtained with a sufficiently small focal interval [9]. In the TIE method, the phase contrasts transferred to the three LTEM images are too weak for the phase distributions to be determined accurately. The phase distributions obtained by the TIE method have been reported to often suffer from noise amplification and image artifacts [10], which makes observing detailed magnetic domain structures difficult. Therefore, to perform precise and accurate focal-series reconstruction, it is necessary to apply a method that can appropriately utilize numerous highly defocused through-focus images. In high-resolution TEM (HRTEM), various focal-series reconstruction methods have been developed in addition to the TIE method. These focal-series reconstruction methods can generally be classified into two types: Fourier analytical reconstruction methods and iterative reconstruction methods. The three-dimensional Fourier filtering method (3DFFM) [11, 12], paraboloid method [13] and real-time wave field reconstruction method [14, 15] are classified as Fourier analytical reconstruction methods. By contrast, the iterative reconstruction methods include the iterative wave function reconstruction (IWFR) method [16], the maximum likelihood method with conjugate gradient method (CG-MAL) [17] and the maximum likelihood method with the steepest descent method (SD-MAL) [18]. Compared with the TIE method, these focal-series reconstruction methods can make use of numerous through-focus images collected under a larger defocus range. In particular, the iterative reconstruction methods can appropriately retrieve low-spatial-frequency components in phase distribution, as previously reported [19], enabling reconstruction of the phase information in a quantitative manner in a wide Fourier spatial frequency range. Therefore, we also expect that they can be used for phase retrieval that utilizes numerous highly defocused through-focus in the LTEM. In this study, we report phase retrieval by the CG-MAL method using through-focus images in the LTEM. As an application example in this study, we applied the method to magnetic bubbles [20, 21]. We evaluated the method using simulated through-focus images of a magnetic bubble, where the images were calculated under the same defocus conditions as those used in an experiment. Subsequently, the phase distribution was retrieved using 32 experimentally obtained through-focus images of magnetic bubbles, which were obtained under a defocus range from approximately −3 mm to 3 mm. The results indicate that the CG-MAL method is particularly promising for detailed analyses of magnetic domain structures using several largely defocused through-focus LTEM images. Theory In this section, we describe the theoretical background and procedures of the CG-MAL method. In the MAL method, phase retrieval is performed by an iterative process. We represent the reconstructed electron wave function after j iterations of the MAL method as ϕx,y(j)(j=0,1,2⋯). The wave function reconstructed by the Fourier analytical reconstruction methods is usually used as the initial wave function ( ϕx,y(0)). In this study, we obtained the initial wave function by 3DFFM [11]. An advantage of using the 3DFFM is that various parameters such as the temporal partial coherence parameter and the linear sample drift are directly measured and evaluated from the 3D Fourier spectrum [22, 23]. Using the known defocus values of the original through-focus images, we can computationally calculate the through-focus images from the reconstructed wave function ( ϕx,y(j)) using Eq. (1) : Ix,y,Δf′(j) (ϕx,y(j))=|ϕx,y(j)⊗tx,y,Δf|2, (1) where x, y and Δf denote the 2D coordinates and the defocus value, respectively, tx,y,Δf denotes the point spread function [18] at each focus position and ⊗ denotes the convolution operator. At this point, we define the difference between the intensities of the original images and computationally calculated images as an error function: E(j)(ϕx,y(j))=1NΔf∑Δf(∑x,y|Ix,y,Δf−Ix,y,Δf′(j)(ϕx,y(j))|2∑x,yIx,y,Δf), (2) where Ix,y,Δf and Ix,y,Δf′(j) are the original and the computationally calculated through-focus images, respectively, NΔf denotes the total number of through-focus images and ∑ denotes a summation operator. The reconstructed wave function for the next iteration ( ϕx,y(j+1)) is calculated such that the current error function ( E(j)) is minimized using convex optimization techniques such as the conjugate gradient method [17] or steepest descent method [18]. These methods are termed the CG-MAL and SD-MAL methods, respectively. The basic idea behind the MAL method is that the error function returns zero when the true wave function is obtained. That is, the true wave function minimizes the error function. The iterative process continues until the error function sufficiently converges. The updated wave function is obtained based on the following equation: ϕx,y(j+1)=ϕx,y(j)−α·dx,y(j)(α>0), (3) where dx,y(j) denotes the search direction and α denotes a feedback parameter that minimizes the error function as much as possible along the search direction. In the case of the SD-MAL method, the search direction ( dx,y(j)) equals the gradient direction of the error function. In the case of the CG-MAL method, the search direction is determined from the gradient direction of the error function and the previous search direction ( dx,y(j−1)) using the Polak–Ribiére formula [17]. The gradient direction of the error function is expressed as gx,y(j)=−1NΔf∑Δf[{(Ix,y,Δf−I′x,y,Δf(j)(ϕx,y(j)))⋅(ϕx,y(j)⊗tx,y,Δf⁎)}⊗tx,y,Δf⁎/∑x,yIx,y,Δf]. (4) If we determine the search direction ( dx,y(j)), then the feedback parameter α is determined by solving the following equation: ∂E(j)(ϕx,y(j)−α·dx,y(j))∂α=0. (5) Although the MAL method can retrieve low-frequency components of the electron phase more precisely via the iterative processes, the method requires a much longer processing time than the Fourier analytical reconstruction methods. In this paper, we developed a software code that utilizes a graphical processing unit (GPU) in order to shorten the processing time of the MAL method. As a result, the processing time of the method was accelerated to ~70 times faster than the processing time with a central processing unit. Materials and methods We observed magnetic bubbles generated in the single-crystal thin films of BaFe10.35Sc1.6Mg0.05O19 [24]. The magnetic bubbles were generated by applying a magnetic field (~200 mT) perpendicular to the sample using an objective lens of a transmission electron microscope (JEOL JEM-2100F, 200 kV). We observed type-I and type-II magnetic bubbles [25] in the thin films. Type-I magnetic bubbles are generally characterized by magnetic domain walls rotationally magnetized in either a clockwise or a counterclockwise direction. Type-II magnetic bubbles are generally characterized by two parallel domain walls with a pair of Bloch lines. The 32 through-focus images of the magnetic bubbles were obtained in a defocus range from −3.36 mm to 3.15 mm with a 210-μm focus interval using an objective mini-lens of the TEM. In this study, the positive value of the defocus axis is set as the under-focus side. The captured through-focus images were aligned using a template-matching method [26] to correct changes in optical parameters such as image rotation, image magnification and image shift. In this scheme, the distance and positional relations between the same types of magnetic bubbles were measured using the template-matching method for each defocused image. The through-focus images were then aligned so that the distance and positional relationship of the bubbles were kept constant between each defocused image. Results and discussion Simulation We evaluated the CG-MAL method using simulated through-focus images of a type-I magnetic bubble. Figure 1a and b shows an assumed phase of electron wave function induced by the magnetic bubble and its line profile along the radial direction, respectively. In this study, the phase shift was assumed to be rotationally symmetric. The maximum phase shift was assumed to be 90° at the center of the magnetic bubble. The phase changes to zero in the form of a half-cycle of a cosine function at the edges of the magnetic bubble. Figure 1c shows the corresponding in-plane magnetization map of the magnetic bubble. In Fig. 1c, the color and brightness correspond to the direction and magnitude of the in-plane magnetization as shown in the color wheel. The in-plane magnetization of the magnetic domain wall was assumed to be localized in the ring-shaped region as indicated by white-colored arrows. The magnetic field in the center of the magnetic bubble is assumed to be upward along the optical axis. By contrast, the magnetic field in the outskirt dark regions is downward along the optical axis. Fig. 1. View largeDownload slide (a) Assumed phase of electron wave function and (b) its line profile along the radial direction of the type-I magnetic bubble. (c) The corresponding in-plane magnetization map of a type-I magnetic bubble; the in-plane magnetization vector is represented by white-colored arrows. The color and brightness in the in-plane magnetization map correspond to the direction and magnitude of the in-plane magnetization, respectively, as shown in the color wheel. Fig. 1. View largeDownload slide (a) Assumed phase of electron wave function and (b) its line profile along the radial direction of the type-I magnetic bubble. (c) The corresponding in-plane magnetization map of a type-I magnetic bubble; the in-plane magnetization vector is represented by white-colored arrows. The color and brightness in the in-plane magnetization map correspond to the direction and magnitude of the in-plane magnetization, respectively, as shown in the color wheel. Figure 2 shows a few LTEM images selected from the 32 simulated through-focus images. These images were calculated under the same defocus conditions as those used in the LTEM experiment. With respect to the in-focus condition, the contrast of the magnetic bubble disappeared, as shown in Fig. 2a, because the type-I magnetic bubble was assumed to be an ideal phase object in the calculation. Figure 2b–e depicts the LTEM images corresponding to the defocus values of ±0.4 mm and ±3.2 mm. By performing large-scale defocusing, we obtained strong contrast in the images of the magnetic bubble, as shown in Fig. 2c and e. The ring-shaped bright and dark contrasts in the LTEM images are understood to be formed by the convergence and divergence of the incident electron beam, respectively. Fig. 2. View largeDownload slide Simulated LTEM images of the type-I magnetic bubble at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The positive value of the defocus axis is set as the under-focus side. Fig. 2. View largeDownload slide Simulated LTEM images of the type-I magnetic bubble at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The positive value of the defocus axis is set as the under-focus side. We performed the phase retrieval using the 32 through-focus images, including the images shown in Fig. 2. Figure 3a shows the phase of the wave function reconstructed by the 3DFFM used as the initial wave function for the CG-MAL method. In Fig. 3a, the low-spatial-frequency components of the phase distribution were not well retrieved because of the effective contrast transfer function [27] of the Fourier analytical reconstruction method. Figure 3b shows the corresponding in-plane magnetization map obtained by differentiating the retrieved phase distribution [28]. The in-plane magnetization map shows several ring artifacts caused by the sub-peaks in the obtained phase distribution. The simulation results indicate that it is not possible to reconstruct the whole wave function under the current experimental conditions by only using such Fourier analytical reconstruction methods. Fig. 3. View largeDownload slide (a), (b) Retrieved phase distribution and corresponding magnetization map obtained using the 3DFFM. The phase distribution and magnetization map obtained by the CG-MAL method after (c), (d) 10 iterations and after (e), (f) 100 iterations. (g) The line profile of the retrieved phase distribution in (e) along the radial direction. Fig. 3. View largeDownload slide (a), (b) Retrieved phase distribution and corresponding magnetization map obtained using the 3DFFM. The phase distribution and magnetization map obtained by the CG-MAL method after (c), (d) 10 iterations and after (e), (f) 100 iterations. (g) The line profile of the retrieved phase distribution in (e) along the radial direction. Figure 3c–f depicts the reconstructed phase distributions and the corresponding in-plane magnetization maps obtained by the CG-MAL method after 10 and 100 iterations, respectively. By repeating the iterative procedures, we could appropriately enhance the low-spatial-frequency components and obtain an in-plane magnetization map equivalent to that assumed in Fig. 1c. Figure 3g shows the line profile of the retrieved phase distribution in Fig. 3e; it demonstrates that the accuracy of the phase retrieval is substantially high compared with that shown in Fig. 1b. In Fig. 3g, the retrieved phase distribution shifts by ~10° in the negative direction because it is theoretically impossible to obtain the offset value of the phase distribution in the focal-series reconstruction methods. Thus, we confirm that the phase distribution of the electron wave is theoretically retrieved by the CG-MAL method. In the CG-MAL method, the detection limit of the phase angle mainly depends on the total number of defocused images, defocus range of the through-focus images, size of the magnetic bubbles and the electron dose. If these parameters are determined, we can estimate the phase-angle determination precision on the basis of the image simulation by considering the electron dose. We have confirmed that the detection limit of the method is approximately a few degrees under the current simulation parameters and the experimental electron dose density (50 electrons/(nm2·frame)). Experimental results We applied the CG-MAL method to the experimentally obtained through-focus images. Figure 4 shows several LTEM images selected from the experimentally obtained 32 through-focus images of the magnetic bubbles. At the in-focus position, the contrast of magnetic bubbles almost disappears, as shown in Fig. 4a. This lack of contrast indicates that the magnetic domain structures of the sample are considered as phase objects. In Fig. 4b and d, we observe a slight contrast of the magnetic bubbles at the defocus values of ±0.4 mm although the contrast is relatively weak. The magnetic bubbles indicated by A and C in Fig. 4b exhibit contrast typical of the type-I magnetic bubbles. The magnetic bubbles indicated by B and D in Fig. 4b exhibit contrast typical of the type-II magnetic bubbles. Conversely, in Fig. 4c and e, we observe stronger Fresnel contrast of the magnetic bubbles compared with those in Fig. 4b and d. Fig. 4. View largeDownload slide Experimentally obtained through-focus images of the magnetic bubbles at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The magnetic bubbles indicated by A and C show the typical contrast of type-I magnetic bubbles. The magnetic bubbles indicated by B and D show the typical contrast of type- II magnetic bubbles. Fig. 4. View largeDownload slide Experimentally obtained through-focus images of the magnetic bubbles at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The magnetic bubbles indicated by A and C show the typical contrast of type-I magnetic bubbles. The magnetic bubbles indicated by B and D show the typical contrast of type- II magnetic bubbles. Figure 5a and b shows the phase distribution and corresponding in-plane magnetization map obtained by the TIE method, respectively. In the TIE method, the retrieved phase and in-plane magnetization vary depending on the three applied defocused images. In the present work, we selected Fig. 4a, b and d as the three images based on the defocus value used in the previously reported TIE method [20]. To eliminate image artifacts, the phase distribution obtained by the TIE method was processed by a low-cut filter that eliminates low-spatial-frequency components with frequencies <2.5 μm−1. The phase distribution and magnetization map appear as relatively well reconstructed, although the magnetization map includes a few noise components and artifacts caused by the bend contour lines. Fig. 5. View largeDownload slide Retrieved phase distribution and corresponding in-plane magnetization map obtained by (a), (b) the TIE method and by (c), (d) the CG-MAL method with 100 iterations. Results (a) and (b) corresponding to the TIE method were obtained using the three LTEM images in Fig. 4(a), (b) and (d). Results (c) and (d) corresponding to the CG-MAL method were obtained by using 32 through-focus images. Fig. 5. View largeDownload slide Retrieved phase distribution and corresponding in-plane magnetization map obtained by (a), (b) the TIE method and by (c), (d) the CG-MAL method with 100 iterations. Results (a) and (b) corresponding to the TIE method were obtained using the three LTEM images in Fig. 4(a), (b) and (d). Results (c) and (d) corresponding to the CG-MAL method were obtained by using 32 through-focus images. Figure 5c and d shows the phase distribution and corresponding in-plane magnetization maps obtained by the CG-MAL method with 100 iterations. The phase distribution and the magnetization map are qualitatively the same as those obtained using the TIE method. Therefore, we experimentally retrieved the phase distribution of the electron wave by the CG-MAL method by using several through-focus images obtained under the defocus range from approximately −3 mm to 3 mm. The current defocus range is several tens of times larger than the previous range studied using the TIE method [20, 21, 24]. Because the CG-MAL method can treat numerous through-focus images in a wide defocus range, the obtained results appear to be more reliable and precise, reducing the noise and artifacts. We believe that the CG-MAL method is superior to the TIE method in this regard, although the CG-MAL method requires a much longer processing time. In Fig. 5d obtained by the CG-MAL method, we observed magnetic domain structures of the magnetic bubbles in detail. For example, the type-I magnetic bubbles indicated by A and C include magnetic domain walls localized in a region with a width of ~109 nm. In addition, the type-II magnetic bubbles indicated by B and D include slightly different detailed magnetic domain structures. In the magnetic bubble indicated by B, both ends of two parallel domain walls are shifted relative to each other. By contrast, both ends of two parallel domain walls face each other in the magnetic bubble indicated by D. In addition, the magnitude of in-plane magnetization is different in the center region of each magnetic bubble. These differences can be caused by differences in the 3D structure of each magnetic bubble. The use of micro-magnetic simulations [29] and the obtained in-plane magnetization map will enable a quantitative discussion of the difference in each 3D magnetic domain structure in the future. Comparison of convergence behavior In this section, we compare the convergence behavior of the CG-MAL method to those of the convergence behaviors of the SD-MAL method and the IWFR method, which correspond to other popular iterative phase retrieval methods used in HRTEM. The IWFR method is based on a Gerchberg–Saxton type algorithm [30] in which a wave function is reconstructed by an iterative 2D Fourier transform and an inverse 2D Fourier transform. These iterative reconstruction methods can also reconstruct the same phase distributions as the CG-MAL method. Figure 6a and b shows the change in the error functions of the IWFR, SD-MAL and CG-MAL methods for the simulation and experiment, as described in Sections 4.1 and 4.2, respectively. Each value of the error functions was calculated based on Eq. (2). The same initial wave function is used for each phase retrieval method; thus, the values of the error functions of the first iteration are identical to each other. In Fig. 6b, the error function of each iteration method does not converge to zero because the experimentally obtained through-focus images include random noise components. The convergence speed of the CG-MAL method exceeds that of the other methods in both the simulation and the experiment. Therefore, we conclude that the CG-MAL method exhibits better convergence behavior than the other reconstruction methods. In addition, the IWFR and SD-MAL methods exhibit similar convergence behaviors. One reason for these similar behaviors is that the Gerchberg–Saxton type algorithm is mathematically equivalent to the steepest descent method [31]. Fig. 6. View largeDownload slide Change in the error functions of the IWFR, SD-MAL and CG-MAL methods for (a) the simulation and (b) the experiment. Fig. 6. View largeDownload slide Change in the error functions of the IWFR, SD-MAL and CG-MAL methods for (a) the simulation and (b) the experiment. In the present study, the processing time of the CG-MAL and SD-MAL methods was ~77 ms per iteration using a GPU (NVIDIA GTX 1070). By contrast, the processing time of the IWFR method using the GPU was ~35 ms. The processing time per iteration of the CG-MAL method is slightly longer than that of the IWFR method ; however, the CG-MAL method enables the converged data to be obtained in much shorter time. In the present study, the image size of the 32 through-focus images corresponds to 1024 × 1024 px, which includes the original image region (512 × 512 px) and the padding regions surrounding the original images with a width of 256 px [18]. Concluding remarks In this study, we reported phase retrieval for magnetic domain structures via the CG-MAL method, which is usually applied to HRTEM observations. As an application example, we evaluated the method using 32 simulated and experimentally obtained through-focus images of magnetic bubbles where the images were collected under a defocus range from approximately −3 mm to 3 mm. As a result, we successfully obtained the magnetic domain structures of the magnetic bubbles. The current defocus range is several tens of times higher than the previous range examined using the TIE method. Furthermore, the CG-MAL method exhibited better convergence behavior than the other iterative reconstruction methods. The method will also be widely and effectively applicable to the observation of magnetic domain structures other than magnetic bubbles. Therefore, the CG-MAL method is promising for the study of detailed magnetic domain structures when several highly defocused through-focus images are used. In the experimental conditions used in the present study, the maximum defocus range of the through-focus images was mainly restricted by the correction accuracy of the optical parameters such as image rotation, image magnification and image shift. Thus, performing further detailed analyses of magnetic domain structures necessitates the development of a TEM system that enables large-scale defocusing with smaller changes in the optical parameters. In the near future, further quantitative observations of magnetic domain structures will be possible through the combination of the TEM system and the phase retrieval method. Funding Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (B) (grant number 25286059). References 1 Nakajima H , Kotani A , Harada K , Ishii Y , and Mori S ( 2016 ) Foucault imaging and small-angle electron diffraction in controlled external magnetic fields . Microscopy 65 : 473 – 478 . 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Phase retrieval using through-focus images in Lorentz transmission electron microscopy

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Abstract

Abstract In this study, we report on phase retrieval by the maximum likelihood method with conjugate gradient method (CG-MAL) using through-focus images in Lorentz transmission electron microscopy (LTEM). The method was evaluated using 32 simulated and experimentally obtained through-focus images of magnetic bubbles; these images were collected under a defocus range from approximately −3 mm to 3 mm. Consequently, we obtained the magnetic domain structures of the magnetic bubbles in both the simulation and the LTEM experiment. Furthermore, the CG-MAL method showed better convergence behavior than other iterative phase retrieval methods. Therefore, the method can also be widely and effectively applied to the observation of magnetic domain structures other than magnetic bubbles when highly defocused through-focus images are used. Lorentz transmission electron microscopy, phase retrieval, focal-series reconstruction, iterative wave function reconstruction, maximum likelihood method, magnetic domain structure Introduction In Lorentz transmission electron microscopy (LTEM) [1], phase retrieval using through-focus images [2] is a powerful technique for observing magnetic domain structures within samples in real space. Although this approach has the advantage of enabling phase retrieval without any additional devices, it also has several disadvantages compared with other phase retrieval techniques. For instance, the time resolution of the focal-series reconstruction methods is usually lower than that of techniques such as differential phase contrast imaging in scanning TEM (DPC-STEM) [3, 4] because the focal-series reconstruction methods usually require multiple defocused images and image processing. More importantly, the quantitative performance of the focal-series reconstruction methods is often inferior to that of off-axis electron holography [5, 6]. Performing the focal-series reconstruction more quantitatively requires utilizing highly defocused images in a wide defocus range because the magnitude of the observed image contrast generally depends on the magnitude of the defocus value. A phase retrieval method that utilizes a transport-of-intensity equation (TIE) [7] (i.e. a TIE method) is the most popular focal-series reconstruction method in LTEM; it generally involves an in-focus image and two defocused images [8]. However, the TIE method has several problems attributed to the fact that the method can appropriately deal with only three LTEM images obtained with a sufficiently small focal interval [9]. In the TIE method, the phase contrasts transferred to the three LTEM images are too weak for the phase distributions to be determined accurately. The phase distributions obtained by the TIE method have been reported to often suffer from noise amplification and image artifacts [10], which makes observing detailed magnetic domain structures difficult. Therefore, to perform precise and accurate focal-series reconstruction, it is necessary to apply a method that can appropriately utilize numerous highly defocused through-focus images. In high-resolution TEM (HRTEM), various focal-series reconstruction methods have been developed in addition to the TIE method. These focal-series reconstruction methods can generally be classified into two types: Fourier analytical reconstruction methods and iterative reconstruction methods. The three-dimensional Fourier filtering method (3DFFM) [11, 12], paraboloid method [13] and real-time wave field reconstruction method [14, 15] are classified as Fourier analytical reconstruction methods. By contrast, the iterative reconstruction methods include the iterative wave function reconstruction (IWFR) method [16], the maximum likelihood method with conjugate gradient method (CG-MAL) [17] and the maximum likelihood method with the steepest descent method (SD-MAL) [18]. Compared with the TIE method, these focal-series reconstruction methods can make use of numerous through-focus images collected under a larger defocus range. In particular, the iterative reconstruction methods can appropriately retrieve low-spatial-frequency components in phase distribution, as previously reported [19], enabling reconstruction of the phase information in a quantitative manner in a wide Fourier spatial frequency range. Therefore, we also expect that they can be used for phase retrieval that utilizes numerous highly defocused through-focus in the LTEM. In this study, we report phase retrieval by the CG-MAL method using through-focus images in the LTEM. As an application example in this study, we applied the method to magnetic bubbles [20, 21]. We evaluated the method using simulated through-focus images of a magnetic bubble, where the images were calculated under the same defocus conditions as those used in an experiment. Subsequently, the phase distribution was retrieved using 32 experimentally obtained through-focus images of magnetic bubbles, which were obtained under a defocus range from approximately −3 mm to 3 mm. The results indicate that the CG-MAL method is particularly promising for detailed analyses of magnetic domain structures using several largely defocused through-focus LTEM images. Theory In this section, we describe the theoretical background and procedures of the CG-MAL method. In the MAL method, phase retrieval is performed by an iterative process. We represent the reconstructed electron wave function after j iterations of the MAL method as ϕx,y(j)(j=0,1,2⋯). The wave function reconstructed by the Fourier analytical reconstruction methods is usually used as the initial wave function ( ϕx,y(0)). In this study, we obtained the initial wave function by 3DFFM [11]. An advantage of using the 3DFFM is that various parameters such as the temporal partial coherence parameter and the linear sample drift are directly measured and evaluated from the 3D Fourier spectrum [22, 23]. Using the known defocus values of the original through-focus images, we can computationally calculate the through-focus images from the reconstructed wave function ( ϕx,y(j)) using Eq. (1) : Ix,y,Δf′(j) (ϕx,y(j))=|ϕx,y(j)⊗tx,y,Δf|2, (1) where x, y and Δf denote the 2D coordinates and the defocus value, respectively, tx,y,Δf denotes the point spread function [18] at each focus position and ⊗ denotes the convolution operator. At this point, we define the difference between the intensities of the original images and computationally calculated images as an error function: E(j)(ϕx,y(j))=1NΔf∑Δf(∑x,y|Ix,y,Δf−Ix,y,Δf′(j)(ϕx,y(j))|2∑x,yIx,y,Δf), (2) where Ix,y,Δf and Ix,y,Δf′(j) are the original and the computationally calculated through-focus images, respectively, NΔf denotes the total number of through-focus images and ∑ denotes a summation operator. The reconstructed wave function for the next iteration ( ϕx,y(j+1)) is calculated such that the current error function ( E(j)) is minimized using convex optimization techniques such as the conjugate gradient method [17] or steepest descent method [18]. These methods are termed the CG-MAL and SD-MAL methods, respectively. The basic idea behind the MAL method is that the error function returns zero when the true wave function is obtained. That is, the true wave function minimizes the error function. The iterative process continues until the error function sufficiently converges. The updated wave function is obtained based on the following equation: ϕx,y(j+1)=ϕx,y(j)−α·dx,y(j)(α>0), (3) where dx,y(j) denotes the search direction and α denotes a feedback parameter that minimizes the error function as much as possible along the search direction. In the case of the SD-MAL method, the search direction ( dx,y(j)) equals the gradient direction of the error function. In the case of the CG-MAL method, the search direction is determined from the gradient direction of the error function and the previous search direction ( dx,y(j−1)) using the Polak–Ribiére formula [17]. The gradient direction of the error function is expressed as gx,y(j)=−1NΔf∑Δf[{(Ix,y,Δf−I′x,y,Δf(j)(ϕx,y(j)))⋅(ϕx,y(j)⊗tx,y,Δf⁎)}⊗tx,y,Δf⁎/∑x,yIx,y,Δf]. (4) If we determine the search direction ( dx,y(j)), then the feedback parameter α is determined by solving the following equation: ∂E(j)(ϕx,y(j)−α·dx,y(j))∂α=0. (5) Although the MAL method can retrieve low-frequency components of the electron phase more precisely via the iterative processes, the method requires a much longer processing time than the Fourier analytical reconstruction methods. In this paper, we developed a software code that utilizes a graphical processing unit (GPU) in order to shorten the processing time of the MAL method. As a result, the processing time of the method was accelerated to ~70 times faster than the processing time with a central processing unit. Materials and methods We observed magnetic bubbles generated in the single-crystal thin films of BaFe10.35Sc1.6Mg0.05O19 [24]. The magnetic bubbles were generated by applying a magnetic field (~200 mT) perpendicular to the sample using an objective lens of a transmission electron microscope (JEOL JEM-2100F, 200 kV). We observed type-I and type-II magnetic bubbles [25] in the thin films. Type-I magnetic bubbles are generally characterized by magnetic domain walls rotationally magnetized in either a clockwise or a counterclockwise direction. Type-II magnetic bubbles are generally characterized by two parallel domain walls with a pair of Bloch lines. The 32 through-focus images of the magnetic bubbles were obtained in a defocus range from −3.36 mm to 3.15 mm with a 210-μm focus interval using an objective mini-lens of the TEM. In this study, the positive value of the defocus axis is set as the under-focus side. The captured through-focus images were aligned using a template-matching method [26] to correct changes in optical parameters such as image rotation, image magnification and image shift. In this scheme, the distance and positional relations between the same types of magnetic bubbles were measured using the template-matching method for each defocused image. The through-focus images were then aligned so that the distance and positional relationship of the bubbles were kept constant between each defocused image. Results and discussion Simulation We evaluated the CG-MAL method using simulated through-focus images of a type-I magnetic bubble. Figure 1a and b shows an assumed phase of electron wave function induced by the magnetic bubble and its line profile along the radial direction, respectively. In this study, the phase shift was assumed to be rotationally symmetric. The maximum phase shift was assumed to be 90° at the center of the magnetic bubble. The phase changes to zero in the form of a half-cycle of a cosine function at the edges of the magnetic bubble. Figure 1c shows the corresponding in-plane magnetization map of the magnetic bubble. In Fig. 1c, the color and brightness correspond to the direction and magnitude of the in-plane magnetization as shown in the color wheel. The in-plane magnetization of the magnetic domain wall was assumed to be localized in the ring-shaped region as indicated by white-colored arrows. The magnetic field in the center of the magnetic bubble is assumed to be upward along the optical axis. By contrast, the magnetic field in the outskirt dark regions is downward along the optical axis. Fig. 1. View largeDownload slide (a) Assumed phase of electron wave function and (b) its line profile along the radial direction of the type-I magnetic bubble. (c) The corresponding in-plane magnetization map of a type-I magnetic bubble; the in-plane magnetization vector is represented by white-colored arrows. The color and brightness in the in-plane magnetization map correspond to the direction and magnitude of the in-plane magnetization, respectively, as shown in the color wheel. Fig. 1. View largeDownload slide (a) Assumed phase of electron wave function and (b) its line profile along the radial direction of the type-I magnetic bubble. (c) The corresponding in-plane magnetization map of a type-I magnetic bubble; the in-plane magnetization vector is represented by white-colored arrows. The color and brightness in the in-plane magnetization map correspond to the direction and magnitude of the in-plane magnetization, respectively, as shown in the color wheel. Figure 2 shows a few LTEM images selected from the 32 simulated through-focus images. These images were calculated under the same defocus conditions as those used in the LTEM experiment. With respect to the in-focus condition, the contrast of the magnetic bubble disappeared, as shown in Fig. 2a, because the type-I magnetic bubble was assumed to be an ideal phase object in the calculation. Figure 2b–e depicts the LTEM images corresponding to the defocus values of ±0.4 mm and ±3.2 mm. By performing large-scale defocusing, we obtained strong contrast in the images of the magnetic bubble, as shown in Fig. 2c and e. The ring-shaped bright and dark contrasts in the LTEM images are understood to be formed by the convergence and divergence of the incident electron beam, respectively. Fig. 2. View largeDownload slide Simulated LTEM images of the type-I magnetic bubble at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The positive value of the defocus axis is set as the under-focus side. Fig. 2. View largeDownload slide Simulated LTEM images of the type-I magnetic bubble at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The positive value of the defocus axis is set as the under-focus side. We performed the phase retrieval using the 32 through-focus images, including the images shown in Fig. 2. Figure 3a shows the phase of the wave function reconstructed by the 3DFFM used as the initial wave function for the CG-MAL method. In Fig. 3a, the low-spatial-frequency components of the phase distribution were not well retrieved because of the effective contrast transfer function [27] of the Fourier analytical reconstruction method. Figure 3b shows the corresponding in-plane magnetization map obtained by differentiating the retrieved phase distribution [28]. The in-plane magnetization map shows several ring artifacts caused by the sub-peaks in the obtained phase distribution. The simulation results indicate that it is not possible to reconstruct the whole wave function under the current experimental conditions by only using such Fourier analytical reconstruction methods. Fig. 3. View largeDownload slide (a), (b) Retrieved phase distribution and corresponding magnetization map obtained using the 3DFFM. The phase distribution and magnetization map obtained by the CG-MAL method after (c), (d) 10 iterations and after (e), (f) 100 iterations. (g) The line profile of the retrieved phase distribution in (e) along the radial direction. Fig. 3. View largeDownload slide (a), (b) Retrieved phase distribution and corresponding magnetization map obtained using the 3DFFM. The phase distribution and magnetization map obtained by the CG-MAL method after (c), (d) 10 iterations and after (e), (f) 100 iterations. (g) The line profile of the retrieved phase distribution in (e) along the radial direction. Figure 3c–f depicts the reconstructed phase distributions and the corresponding in-plane magnetization maps obtained by the CG-MAL method after 10 and 100 iterations, respectively. By repeating the iterative procedures, we could appropriately enhance the low-spatial-frequency components and obtain an in-plane magnetization map equivalent to that assumed in Fig. 1c. Figure 3g shows the line profile of the retrieved phase distribution in Fig. 3e; it demonstrates that the accuracy of the phase retrieval is substantially high compared with that shown in Fig. 1b. In Fig. 3g, the retrieved phase distribution shifts by ~10° in the negative direction because it is theoretically impossible to obtain the offset value of the phase distribution in the focal-series reconstruction methods. Thus, we confirm that the phase distribution of the electron wave is theoretically retrieved by the CG-MAL method. In the CG-MAL method, the detection limit of the phase angle mainly depends on the total number of defocused images, defocus range of the through-focus images, size of the magnetic bubbles and the electron dose. If these parameters are determined, we can estimate the phase-angle determination precision on the basis of the image simulation by considering the electron dose. We have confirmed that the detection limit of the method is approximately a few degrees under the current simulation parameters and the experimental electron dose density (50 electrons/(nm2·frame)). Experimental results We applied the CG-MAL method to the experimentally obtained through-focus images. Figure 4 shows several LTEM images selected from the experimentally obtained 32 through-focus images of the magnetic bubbles. At the in-focus position, the contrast of magnetic bubbles almost disappears, as shown in Fig. 4a. This lack of contrast indicates that the magnetic domain structures of the sample are considered as phase objects. In Fig. 4b and d, we observe a slight contrast of the magnetic bubbles at the defocus values of ±0.4 mm although the contrast is relatively weak. The magnetic bubbles indicated by A and C in Fig. 4b exhibit contrast typical of the type-I magnetic bubbles. The magnetic bubbles indicated by B and D in Fig. 4b exhibit contrast typical of the type-II magnetic bubbles. Conversely, in Fig. 4c and e, we observe stronger Fresnel contrast of the magnetic bubbles compared with those in Fig. 4b and d. Fig. 4. View largeDownload slide Experimentally obtained through-focus images of the magnetic bubbles at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The magnetic bubbles indicated by A and C show the typical contrast of type-I magnetic bubbles. The magnetic bubbles indicated by B and D show the typical contrast of type- II magnetic bubbles. Fig. 4. View largeDownload slide Experimentally obtained through-focus images of the magnetic bubbles at defocus values corresponding to (a) 0.0 mm, (b) 0.4 mm, (c) 3.2 mm, (d) −0.4 mm and (e) −3.2 mm. The magnetic bubbles indicated by A and C show the typical contrast of type-I magnetic bubbles. The magnetic bubbles indicated by B and D show the typical contrast of type- II magnetic bubbles. Figure 5a and b shows the phase distribution and corresponding in-plane magnetization map obtained by the TIE method, respectively. In the TIE method, the retrieved phase and in-plane magnetization vary depending on the three applied defocused images. In the present work, we selected Fig. 4a, b and d as the three images based on the defocus value used in the previously reported TIE method [20]. To eliminate image artifacts, the phase distribution obtained by the TIE method was processed by a low-cut filter that eliminates low-spatial-frequency components with frequencies <2.5 μm−1. The phase distribution and magnetization map appear as relatively well reconstructed, although the magnetization map includes a few noise components and artifacts caused by the bend contour lines. Fig. 5. View largeDownload slide Retrieved phase distribution and corresponding in-plane magnetization map obtained by (a), (b) the TIE method and by (c), (d) the CG-MAL method with 100 iterations. Results (a) and (b) corresponding to the TIE method were obtained using the three LTEM images in Fig. 4(a), (b) and (d). Results (c) and (d) corresponding to the CG-MAL method were obtained by using 32 through-focus images. Fig. 5. View largeDownload slide Retrieved phase distribution and corresponding in-plane magnetization map obtained by (a), (b) the TIE method and by (c), (d) the CG-MAL method with 100 iterations. Results (a) and (b) corresponding to the TIE method were obtained using the three LTEM images in Fig. 4(a), (b) and (d). Results (c) and (d) corresponding to the CG-MAL method were obtained by using 32 through-focus images. Figure 5c and d shows the phase distribution and corresponding in-plane magnetization maps obtained by the CG-MAL method with 100 iterations. The phase distribution and the magnetization map are qualitatively the same as those obtained using the TIE method. Therefore, we experimentally retrieved the phase distribution of the electron wave by the CG-MAL method by using several through-focus images obtained under the defocus range from approximately −3 mm to 3 mm. The current defocus range is several tens of times larger than the previous range studied using the TIE method [20, 21, 24]. Because the CG-MAL method can treat numerous through-focus images in a wide defocus range, the obtained results appear to be more reliable and precise, reducing the noise and artifacts. We believe that the CG-MAL method is superior to the TIE method in this regard, although the CG-MAL method requires a much longer processing time. In Fig. 5d obtained by the CG-MAL method, we observed magnetic domain structures of the magnetic bubbles in detail. For example, the type-I magnetic bubbles indicated by A and C include magnetic domain walls localized in a region with a width of ~109 nm. In addition, the type-II magnetic bubbles indicated by B and D include slightly different detailed magnetic domain structures. In the magnetic bubble indicated by B, both ends of two parallel domain walls are shifted relative to each other. By contrast, both ends of two parallel domain walls face each other in the magnetic bubble indicated by D. In addition, the magnitude of in-plane magnetization is different in the center region of each magnetic bubble. These differences can be caused by differences in the 3D structure of each magnetic bubble. The use of micro-magnetic simulations [29] and the obtained in-plane magnetization map will enable a quantitative discussion of the difference in each 3D magnetic domain structure in the future. Comparison of convergence behavior In this section, we compare the convergence behavior of the CG-MAL method to those of the convergence behaviors of the SD-MAL method and the IWFR method, which correspond to other popular iterative phase retrieval methods used in HRTEM. The IWFR method is based on a Gerchberg–Saxton type algorithm [30] in which a wave function is reconstructed by an iterative 2D Fourier transform and an inverse 2D Fourier transform. These iterative reconstruction methods can also reconstruct the same phase distributions as the CG-MAL method. Figure 6a and b shows the change in the error functions of the IWFR, SD-MAL and CG-MAL methods for the simulation and experiment, as described in Sections 4.1 and 4.2, respectively. Each value of the error functions was calculated based on Eq. (2). The same initial wave function is used for each phase retrieval method; thus, the values of the error functions of the first iteration are identical to each other. In Fig. 6b, the error function of each iteration method does not converge to zero because the experimentally obtained through-focus images include random noise components. The convergence speed of the CG-MAL method exceeds that of the other methods in both the simulation and the experiment. Therefore, we conclude that the CG-MAL method exhibits better convergence behavior than the other reconstruction methods. In addition, the IWFR and SD-MAL methods exhibit similar convergence behaviors. One reason for these similar behaviors is that the Gerchberg–Saxton type algorithm is mathematically equivalent to the steepest descent method [31]. Fig. 6. View largeDownload slide Change in the error functions of the IWFR, SD-MAL and CG-MAL methods for (a) the simulation and (b) the experiment. Fig. 6. View largeDownload slide Change in the error functions of the IWFR, SD-MAL and CG-MAL methods for (a) the simulation and (b) the experiment. In the present study, the processing time of the CG-MAL and SD-MAL methods was ~77 ms per iteration using a GPU (NVIDIA GTX 1070). By contrast, the processing time of the IWFR method using the GPU was ~35 ms. The processing time per iteration of the CG-MAL method is slightly longer than that of the IWFR method ; however, the CG-MAL method enables the converged data to be obtained in much shorter time. In the present study, the image size of the 32 through-focus images corresponds to 1024 × 1024 px, which includes the original image region (512 × 512 px) and the padding regions surrounding the original images with a width of 256 px [18]. Concluding remarks In this study, we reported phase retrieval for magnetic domain structures via the CG-MAL method, which is usually applied to HRTEM observations. As an application example, we evaluated the method using 32 simulated and experimentally obtained through-focus images of magnetic bubbles where the images were collected under a defocus range from approximately −3 mm to 3 mm. As a result, we successfully obtained the magnetic domain structures of the magnetic bubbles. The current defocus range is several tens of times higher than the previous range examined using the TIE method. Furthermore, the CG-MAL method exhibited better convergence behavior than the other iterative reconstruction methods. The method will also be widely and effectively applicable to the observation of magnetic domain structures other than magnetic bubbles. Therefore, the CG-MAL method is promising for the study of detailed magnetic domain structures when several highly defocused through-focus images are used. In the experimental conditions used in the present study, the maximum defocus range of the through-focus images was mainly restricted by the correction accuracy of the optical parameters such as image rotation, image magnification and image shift. Thus, performing further detailed analyses of magnetic domain structures necessitates the development of a TEM system that enables large-scale defocusing with smaller changes in the optical parameters. In the near future, further quantitative observations of magnetic domain structures will be possible through the combination of the TEM system and the phase retrieval method. Funding Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (B) (grant number 25286059). References 1 Nakajima H , Kotani A , Harada K , Ishii Y , and Mori S ( 2016 ) Foucault imaging and small-angle electron diffraction in controlled external magnetic fields . Microscopy 65 : 473 – 478 . 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MicroscopyOxford University Press

Published: Mar 26, 2018

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