TY - JOUR
AU1 - Yamasaki,, Jun
AB - Abstract In electron diffractive imaging, the phase image of a sample is reconstructed from its diffraction intensity through iterative calculations. The principle of this method is based on the Fourier transform relation between the real-space wave field transmitted by the sample and its Fraunhofer diffraction wave field. Since Gerchberg’s experimental work in 1972, various advancements have been achieved, which have substantially improved the quality of the reconstructed phase images and extended the applicable range of the method. In this review article, the principle of diffractive imaging, various experimental processes using electron beams and application to specific samples are explained in detail. diffractive imaging, phase imaging, electron diffraction, electron beam Introduction Structure analysis in a transmission electron microscope (TEM) is conducted using various imaging/diffraction methods, where the electrons transmitted through and/or scattered by a material are recorded by different means depending on the respective method. The complete scattering information is included in the electron wave field (WF) below the material and its Fraunhofer diffraction WF, both of which are generally represented by complex functions. However, as is well known as the phase problem, only the amplitudes of the WF are recorded by the detector. Thus, to observe the entire complex WF, various phase-imaging methods, including electron holography, have been developed and applied to biological and material sciences. Assuming coherent illumination, determining the phase of the Fraunhofer diffraction WF is equivalent to knowing the phase of the transmitted WF based on the Fourier transform (FT) relation between the WFs. Phase determinations for diffraction spots are well established, especially in X-ray crystallography, to determine the unit-cell structure of a crystal, that is, to determine the average of the periodic structure. Meanwhile, information pertaining to nonperiodic structures appears between Bragg spots. Thus, for phase imaging of a localized structure, the phase of the diffraction pattern must be determined not only at the Bragg spots but also over the entire angular region between the spots. ‘Diffractive imaging’ is a method for retrieving the phase of diffraction patterns based on certain constraints in real space. The resultant phase-retrieved diffraction WF is Fourier-transformed to obtain the real-space WF, from which the phase image of the sample is obtained. Because of the lack of high-precision lenses, this method has been effectively used for X-ray imaging. Since a high degree of coherence is required in the illumination beam as explained later, the use of a synchrotron source or an X-ray free-electron laser (XFEL) is essential; this technique is called coherent X-ray diffractive imaging (CXDI). However, electron diffractive imaging (EDI) can be conducted in a conventional medium-/low-voltage electron microscope equipped with a field-emission gun (FEG). Compared to X-rays, the electron beams in TEMs can be freely controlled using lenses located above and below the sample plane, and in some cases, electron micrographs can be utilized to impose a constraint to determine the phase. Therefore, there are several effective combinations between the electron optics and the ways in which the constraints are imposed, which are selected depending on the target material. In this review article, various EDI methods conducted to make the experimental data suitable for various samples and purposes are explained in detail from the viewpoint of classification of the constraints in real space. Basic knowledge Let us consider a situation where a diffraction pattern of a material is recorded using an |${\rm{N}} \times {\rm{N}}$| detector, as shown in Fig. 1a. Although the recordable information only includes the intensity composed of N2 real numbers, the original reciprocal-space WF is a complex-numbered function composed of 2N2 parameters. The discrete FT of the |${\rm{N}} \times {\rm{N}}$| array of complex numbers results in another |${\rm{N}} \times {\rm{N}}$| array of complex numbers that represents the real-space WF transmitted through the material (Fig. 1a). Thus, the total number of parameters involved in this system is 4N2, which can be determined if 4N2 equations are known. The 4N2 parameters are not completely independent, but are related through the FT relation. If the complex values at the |$\left( {m,n} \right)$| pixel in real space and at the |$(h,k)$| pixel in reciprocal space are denoted by |${\psi _{mn}} = \psi _{mn}^R + i \psi _{mn}^I$| and |${{{\varPsi }}_{hk}} = {{\varPsi }}_{hk}^R + i {{\varPsi }}_{hk}^I$|⁠, respectively, the discrete FT can be described as $$\begin{equation}{{{\varPsi }}_{hk}} = \mathop \sum \limits_{m,n = 0}^{N - 1} \left( {\psi _{mn}^R + i \psi _{mn}^I} \right){\rm{exp}}\left[ { - i{{2\pi } \over N}\left( {mh + nk} \right)} \right]. \end{equation}$$(1) Fig. 1. Open in new tabDownload slide Classification of various real-space constraints in EDI. (a) Image intensity is used as the real-space constraint in the GS algorithm for phase retrieval from the diffraction intensity (two-intensity measurements). (b) The area surrounding an isolated material is used as the real-space constraint in the ER and HIO algorithms (single-intensity measurements). (c) The dark area formed by an ultrasmall selector aperture is used as the real-space constraint for the phase imaging of not only isolated but also extended materials [29]. (d) Both dark area and BF-TEM image intensities inside a selector aperture are used as the real-space constraints for phase imaging from a small-angle SAD pattern (left panel) [3]. (e) The overlapping areas between adjacent illuminated areas are used as a type of real-space constraint in ptychograpic EDI [36, 45]. Images (c) and (d) are reproduced with permission from the AIP Publishing. EDI, electron diffractive imaging; GS, Gerchberg–Saxton; ER, error reduction; HIO, hybrid input–output; BF-TEM, bright-field transmission electron microscopy; SAD, selected-area diffraction. Fig. 1. Open in new tabDownload slide Classification of various real-space constraints in EDI. (a) Image intensity is used as the real-space constraint in the GS algorithm for phase retrieval from the diffraction intensity (two-intensity measurements). (b) The area surrounding an isolated material is used as the real-space constraint in the ER and HIO algorithms (single-intensity measurements). (c) The dark area formed by an ultrasmall selector aperture is used as the real-space constraint for the phase imaging of not only isolated but also extended materials [29]. (d) Both dark area and BF-TEM image intensities inside a selector aperture are used as the real-space constraints for phase imaging from a small-angle SAD pattern (left panel) [3]. (e) The overlapping areas between adjacent illuminated areas are used as a type of real-space constraint in ptychograpic EDI [36, 45]. Images (c) and (d) are reproduced with permission from the AIP Publishing. EDI, electron diffractive imaging; GS, Gerchberg–Saxton; ER, error reduction; HIO, hybrid input–output; BF-TEM, bright-field transmission electron microscopy; SAD, selected-area diffraction. Thus, the following two equations hold for the |$\left( {h,k} \right)$| pixel in reciprocal space: $$\begin{equation}\varPsi _{hk}^R = \mathop \sum \limits_{m,n = 0}^{N - 1} \psi _{mn}^R{\rm{cos}}\left[ {{{2\pi } \over N}\left( {mh + nk} \right)} \right] + \psi _{mn}^I{\rm{sin}}\left[ {{{2\pi } \over N}\left( {mh + nk} \right)} \right], \end{equation}$$(2) $$\begin{equation}\varPsi _{hk}^I = \mathop \sum \limits_{m,n = 0}^{N - 1} \psi _{mn}^I{\rm{cos}}\left[ {{{2\pi } \over N}\left( {mh + nk} \right)} \right] - \psi _{mn}^R{\rm{sin}}\left[ {{{2\pi } \over N}\left( {mh + nk} \right)} \right]. \end{equation}$$(3) As a similar equation pair holds for each pixel, the FT relation leads to 2N2 equations. Similarly, the same number of equations of the same form holds in real space. As they are derived by solving Eqs. (2) and (3) for |$\psi _{mn}^R$| and|$ \psi _{mn}^I$|⁠, the number of independent equations remains 2N2. The square root of the diffraction intensity is the amplitude of the reciprocal-space WF. The recorded diffraction intensity |$I_{hk}^{diff}$| leads to N2 equations in the form of |$I_{hk}^{diff} = {{\varPsi }}{_{hk}^{R^2}} + {{\varPsi }}{_{hk}^{I^2}}$|⁠. This is called the ‘reciprocal-space constraint’, which is used in the phase retrieval process of diffractive imaging. Summing up the FT equations of 2N2 and the reciprocal-space constraints of N2, the total number of equations is 3N2, which is still insufficient for determining the 4N2 parameters. This lack of a sufficient number of equations originally comes from the phase problem, that is, the loss of N2 phase information in the diffraction data. The basic strategy of diffractive imaging is to compensate for the lack of phase information by imposing constraints in real space (referred to as real-space constraints). As explained in the following sections, there are several ways of imposing real-space constraints. Phase imaging based on two-intensity measurements: Gerchberg–Saxton algorithm For microscopists, the most natural means of imposing additional constraints in real space is to use the micrograph intensity corresponding to the diffraction pattern [1–3]. In medium-/low-resolution TEM images, disturbances induced by lens aberrations may be almost insignificant. In such a case, the image intensity leads to N2 equations in the form of |$I_{mn}^{image} = {{\varPsi }}_{mn}^{R\enspace 2} + {{\varPsi }}_{mn}^{I\enspace 2}$|⁠. Because of this real-space constraint, the number of equations reaches 4N2, which is sufficient, in principle, for determining the complex WFs in both spaces. The point is how to actually solve 4N2 nonlinear simultaneous equations. Precisely, because of the noise included in the measured data, an exact solution that satisfies all the equations does not exist. Thus, the question arises how to determine the optimal solution with the least violations of the real- and reciprocal-space constraints. It is known that the optimal solution can be found through iterative calculations of FT and inverse FT with real- and reciprocal-space constraints imposed in alternation, as schematically shown in Fig. 2. This method is generally referred to as ‘iterative FT phase retrieval’, which is widely used as a basic strategy in diffractive imaging using various real-space constraints. Fig. 2. Open in new tabDownload slide Diagram showing process for iterative Fourier-transform phase retrieval. Refer to text for details. Fig. 2. Open in new tabDownload slide Diagram showing process for iterative Fourier-transform phase retrieval. Refer to text for details. Gerchberg and Saxton proposed a specific procedure in case where the image intensity is used for the real-space constraint, called the ‘Gerchberg–Saxton algorithm’ (GS algorithm) [1, 2]. As mentioned earlier, the amplitude of the reciprocal WF is given by the square root of the measured diffraction intensity |${I^{diff}}$|⁠, where |${{{\boldsymbol q}}}$| represents a two-dimensional (2D) reciprocal vector instead of the |$\left( {h,k} \right)$| notation. The unknown phase distribution is tentatively given by |$\Theta {^{\prime}_0}\left( {{{\boldsymbol q}}} \right)$| which comprises random numbers between 0 and 2π. Thus, the initial WF in reciprocal space is prepared by combining these terms as |$\varPsi {^{\prime}_0}\left( {{{\boldsymbol q}}} \right) = \sqrt {{I^{diff}}\left( {{{\boldsymbol q}}} \right)} {\rm{ exp}}\left[ {i\Theta {^{\prime}_0}\left( {{{\boldsymbol q}}} \right)} \right]$|⁠. The inverse FT of |$\varPsi {^{\prime}_0}\left( {{{\boldsymbol q}}} \right)$| results in the initial estimate of the real-space WF, |$\psi {^{\prime}_0}\left( {{{\boldsymbol r}}} \right) = \left| {\psi {^{\prime}_0}\left( {{{\boldsymbol r}}} \right)} \right|exp\left[ {i\theta {^{\prime}_0}\left( {{{\boldsymbol r}}} \right)} \right]$|⁠, where |${{{\boldsymbol r}}}$| represents a 2D position vector. Then, |$\psi {^{\prime}_0}\left( {{{\boldsymbol r}}} \right)$| is modified to |$\; {\psi _1}\left( {{{\boldsymbol r}}} \right) = \sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} {\rm{ exp}}\left[ {i{\theta _1}\left( {{{\boldsymbol r}}} \right)} \right]$| by replacing the amplitude with the square root of the image intensity |${I^{image}}\left( {{{\boldsymbol r}}} \right)$| as the real-space constraint, while the phase is preserved as |${\theta _1}\left( {{{\boldsymbol r}}} \right) = {\theta^{\prime}_0}\left( {{{\boldsymbol r}}} \right)$|⁠. The FT of |${\psi _1}\left( {{{\boldsymbol r}}} \right)$| is considered a tentative reciprocal-space WF, |$\varPsi {_1}\left( {{{\boldsymbol q}}} \right) = \left| \varPsi_1 \left({{{\boldsymbol q}}}\right)\right| {\rm{ exp}}\left[ {i\Theta_1\left({{{\boldsymbol q}}} \right)} \right]$|⁠. Then, |${\varPsi _1}\left( {{{\boldsymbol q}}} \right)$| is modified to |$\varPsi {^{\prime}_1}\left( {{{\boldsymbol q}}} \right) = \sqrt {{I^{diff}}\left( {{{\boldsymbol q}}} \right)} {\rm{ exp}}\left[ {i\Theta {^{\prime}_1}\left( {{{\boldsymbol q}}} \right)} \right]$| by replacing the amplitude with the square root of the diffraction intensity |${I^{diff}}\left( {{{\boldsymbol q}}} \right)$| as the reciprocal-space constraint, while the phase is preserved as |${\Theta^{\prime}_1}\left( {{{\boldsymbol q}}} \right) = {\Theta _1}\left( {{{\boldsymbol q}}} \right)$|⁠. In this manner, the FT and the inverse FT are executed repeatedly while applying both the constraints, until the updated WF converges to the optimal solution. The operations of the nth loop based on the reciprocal- and real-space constraints are rewritten as follows: $$\begin{equation}\varPsi {^{\prime}_n}\left( {{{\boldsymbol q}}} \right) = \sqrt {{I^{diff}}\left( {{{\boldsymbol q}}} \right)} {\rm{ exp}}\left[ {i{\Theta _n}\left( {{{\boldsymbol q}}} \right)} \right] = \sqrt {{I^{diff}}\left( {{{\boldsymbol q}}} \right)} {{{\varPsi _n}\left( {{{\boldsymbol q}}} \right)} \over {\left| {{\varPsi _n}\left( {{{\boldsymbol q}}} \right)} \right|}}\end{equation}$$(4) $$\begin{equation}{\psi _{n + 1}}\left( {{{\boldsymbol r}}} \right) = \sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} {\rm{ exp}}\left[ {i\theta {{^{\prime}}_n}\left( {{{\boldsymbol r}}} \right)} \right] = \sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} {{\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \over {\left| {\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \right|}}\end{equation}$$(5) This procedure using the GS algorithm is referred to as diffractive imaging based on ‘two-intensity measurements’ because it is based on the experimental data measured in both real and reciprocal spaces. Using the GS algorithm, Gerchberg demonstrated experimental phase imaging of an organic crystalline structure [4]. Figure 3a and b shows a selected-area diffraction (SAD) pattern and in-focus TEM image of a small region in the selected area, taken from a negatively stained crystal of beef liver catalase. Considering the field of view (FOV) of Fig. 3b (estimated as ∼7 nm), the image resolution is low enough that the in-focus image can be approximated to an amplitude image rather than a phase-contrast image. Figure 3c shows the amplitude image created by taking the square root of the TEM image’s intensity averaged over the selected area. Using the ‘amplitude image’ shown in Fig. 3c as the real-space constraint and the square root of the diffraction intensity as the reciprocal-space constraint, the phase image shown in Fig. 3d was successfully reconstructed. Because the obtained result is an averaged phase image for the selected area, where several thousands of unit cells are included, this result may not be considered imaging of local structures, in reality. However, from the viewpoint that phase imaging based on a diffraction pattern was achieved for the first time, it can be said that this represents a historical and epoch-making achievement. Phase imaging based on single-intensity measurements: oversampling Basic principle Phase retrieval based on two-intensity measurements is inconvenient for diffractive imaging by X-rays, for which, unlike the electron beam, there is no high-quality lens to form an image. Therefore, an alternative procedure based only on diffraction measurements, that is, on ‘single-intensity measurements’, has been devised [5, 6]. This procedure that does not rely on image intensity is not only convenient for CXDI but also useful for atomic-resolution EDI. This is because high-resolution TEM (HRTEM) images are substantially influenced by lens aberration; thus, unlike medium-resolution images, such as that shown in Fig. 3, their intensity does not directly correspond to the WF amplitude. However, in exchange for avoiding microscope image acquisition, the phase imaging target of this procedure is limited to materials isolated at the center of the FOV such as a nanoparticle or nanotube. This is because the free space surrounding such a compact material is used as a real-space constraint, instead of the TEM image intensity of the GS algorithm. Fig. 3. Open in new tabDownload slide Historically first result of phase imaging based on a diffraction pattern. (a) SAD pattern, (b) TEM image of a small area, (c) averaged amplitude image, and (d) reconstructed phase image of a negatively stained crystal of beef liver catalase [4]. Reprinted with permission from the Springer Nature (https://www.nature.com/). SAD, selected-area diffraction; TEM, transmission electron microscopy. Fig. 3. Open in new tabDownload slide Historically first result of phase imaging based on a diffraction pattern. (a) SAD pattern, (b) TEM image of a small area, (c) averaged amplitude image, and (d) reconstructed phase image of a negatively stained crystal of beef liver catalase [4]. Reprinted with permission from the Springer Nature (https://www.nature.com/). SAD, selected-area diffraction; TEM, transmission electron microscopy. Consider a simple hypothetical situation shown in Fig. 1b, where a square particle expressed by |${\rm{N}} \times {\rm{N}}$| pixels placed at the center of an |${\rm{M}} \times {\rm{M}}$| region (⁠|${\rm{M}} \gt {\rm{N}}$|⁠) is surrounded by vacuum. Similarly to the case of Fig. 1a, the number of equations required for the real-space constraint is calculated as M2. The dimensions of the surrounding area are |${M^2} - {N^2}$|⁠, where the WF is the illumination WF itself, that is, normalized to the real part of 1 and the imaginary part of 0. As these two parameters are determined for each pixel in the surrounding area, |$2\left( {{M^2} - {N^2}} \right)$| equations are given as a real-space constraint to solve the problem. Thus, in principle, as a result of solving |$2\left( {{M^2} - {N^2}} \right) \ge {M^2}$|⁠, the WF can be determined when |${\rm{M}} \ge {\rm{N}}\sqrt 2 $|⁠. In experiments, this condition is satisfied by oversampling the diffraction intensity more than |$\sqrt 2 $| times finer than Fig. 1a, as shown in Fig. 1b. For ease of understanding, the principle of oversampling was explained above using a hypothetical particle of known size and known shape, because this information is important for determining the required degree of oversampling. However, this information is unknown during an experiment because an image of the particle has not yet been reconstructed. Therefore, in some cases, the sample size/shape are roughly estimated based on a low-resolution TEM image [7, 8], scanning electron microscopy (SEM) image [9], or the autocorrelation function obtained by performing FT on the diffraction pattern [10]. As no material is allowed to exist in the surrounding reference region, the central region, called the ‘support’, is set to be larger than the pre-estimated sample size to ensure that the sample is included. The ratio for oversampling the 2D diffraction pattern, |${\left( {M/N} \right)^2}$|⁠, is called the ‘oversampling ratio’, and is equal to the area ratio of the reconstruction FOV and support. Considering the required condition |${\rm{M}} \ge {\rm{N}}\sqrt 2 $|⁠, the ratio of 2 should, in principle, be sufficient to determine all parameters. In practice, however, as with the reconstruction using the GS algorithm, an exact solution to the simultaneous equations does not exist due to the noise included in the experimental data. Increasing the number of equations should be an effective means for reducing the influence of the noise and decreasing the gap between the exact and optimal solutions. In addition, large count numbers at the direct spot tend to be lost because of the saturation of the detector or the effect of the beam stopper [7–15]. Thus, experimentally, the oversampling ratio is generally set to exceed two. As explained in the previous section, the real-space constraint has been given directly from the TEM image in the two-intensity measurements. On the other hand, the real-space constraint in single-intensity measurements is given based on the information about the sample size/shape only in the indirect manner mentioned earlier. This results in frequent occurrence of stagnation in the iteration procedure. Therefore, additional constraints, approximations, or algorithm improvements are required for successful phase retrievals, as introduced in the following sections. Another important experimental consideration is the high degree of coherence of the illumination beam. So far, the principle of the method has been explained assuming a perfectly coherent illumination emitted from an ideal point source. However, a practical extended source results in the spread of the illumination angle at each position in the sample plane, resulting in blurring of the diffraction pattern. According to the van Cittert–Zernike theorem, an increase in the angler spread is equivalent to reduction in the spatial coherence of the illumination. Such partial spatial coherence is described by the attenuation of the coherence function, which describes the degree of coherence between any two points on the sample plane as a function of their separation distance. The standard deviation of the attenuation curve is used as a numerical indicator of the partial spatial coherence, and is known as the spatial coherence length. As the matter of course, such blurring in the diffraction pattern induces inconsistencies between the constraints in each space connected by the FT, resulting in artifacts in the reconstructed results [16, 17]. It has been pointed out that diffractive imaging requires a spatial coherence length of at least twice the size of the scattering material to be imaged [18]. Thus, the method is generally referred to as ‘coherent’ diffractive imaging for X-rays and sometimes for electron beams as well. As mentioned earlier, the former requires a synchrotron or XFEL source, while the latter requires an FEG instead of a thermal gun. Reconstructions of weak-phase objects The simplest means for producing additional constraints in real space is the weak-phase object approximation (WPOA), under which the transmitted WF and its FT are written as $$\begin{equation}{\psi _w}\left( {{{\boldsymbol r}}} \right) = {\rm{exp}}\left[ { - i\sigma {V_p}\left( {{{\boldsymbol r}}} \right)} \right] \cong 1 - i\sigma {V_p}\left( {{{\boldsymbol r}}} \right)\end{equation}$$(6) and $$\begin{equation}{\varPsi _w}\left( {{{\boldsymbol q}}} \right) = \delta \left( {{{\boldsymbol q}}} \right) - i\sigma \mathcal{F}\left[ {{V_p}} \right]\left( {{{\boldsymbol q}}} \right) = \delta \left( {{{\boldsymbol q}}} \right) - i\sigma S\left( {{{\boldsymbol q}}} \right)\end{equation}$$(7) respectively. Here, |$\mathcal{F}$| represents the 2D FT and |$\delta \left( {{{\boldsymbol q}}} \right)$| indicates the delta function. |$\sigma$| is |$2\pi /\lambda E$|⁠, where |$\lambda $| and |$E$| are the wavelength and beam energy, respectively. |${V_p}\left( {{{\boldsymbol r}}} \right)$| and |$S\left( {{{\boldsymbol q}}} \right)$| are the projected potential (negative value) and structure factor in the plane perpendicular to the incident beam. As mentioned earlier, in many experiments, the direct spot is not recorded accurately, or at all, because of intensity saturation or a beam stopper. In such cases, direct spot information is not incorporated into the phase retrieval procedure. When the information at q = 0 is omitted from Eqs. (6) and (7), the real-space WF comprises pure imaginary numbers |${\psi _w}\left( {{{\boldsymbol r}}} \right) = - {\rm{i}}\sigma {V_p}\left( {{{\boldsymbol r}}} \right) = {\mathcal{F}^{ - 1}}\left[ { - i\sigma S\left( {{{\boldsymbol q}}} \right)} \right]$|⁠. The corresponding diffraction intensity, excluding the direct spot, is given as $$\begin{equation}I_w^{diff}\left( {{{\boldsymbol q}}} \right) = {\left| { - i\sigma S\left( {{{\boldsymbol q}}} \right)} \right|^2} = {\left| { - \sigma S\left( {{{\boldsymbol q}}} \right)} \right|^2} \left( {{{{\boldsymbol q}}} \ne 0} \right)\end{equation}$$(8) Equation (8) indicates that both |$ - {\rm{i}}\sigma {V_p}\left( {{{\boldsymbol r}}} \right) = {\mathcal{F}^{ - 1}}\left[ { - i\sigma S\left( {{{\boldsymbol q}}} \right)} \right]$| (pure imaginary numbers) and |$ - \sigma {V_p}\left( {{{\boldsymbol r}}} \right) = {\mathcal{F}^{ - 1}}\left[ { - \sigma S\left( {{{\boldsymbol q}}} \right)} \right]$| (pure real numbers) satisfy the reciprocal-space constraint given by |$I_w^{diff}\left( {{{\boldsymbol q}}} \right)$|⁠. Therefore, under the constraint to pure real (or imaginary) numbers in real space, the latter (former) solution is automatically derived by the iterative phase-retrieval calculation. Such a real-numbered solution, |$ - \sigma {V_p}\left( {{{\boldsymbol r}}} \right)$|⁠, is identical to not the WF itself but its phase image, which is proportional to the projected potential |${V_p}\left( {{{\boldsymbol r}}} \right)$| under WPOA. Considering |${V_p}\left( {{{\boldsymbol r}}} \right)$| for the incident electrons being negative in materials, a more effective real-space constraint is confining the solution |$ - \sigma {V_p}\left( {{{\boldsymbol r}}} \right)$| to the non-negative real numbers inside the support and a zero value outside the support. This is called the ‘non-negativity constraint’, which is widely used in reconstructing weak-phase objects, as introduced later. As an extension of the GS algorithm the error reduction (ER) algorithm was devised for the phase retrieval procedure based on a single-intensity measurement [5]. In the ER algorithm, the real-space constraint of Eq. (5) is changed to $$\begin{equation}{\psi _{n + 1}}\left( {{{\boldsymbol r}}} \right) = \left\{ \begin{array}{ll} & \psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right) & {\boldsymbol r} \in S \\ & 0 & {\boldsymbol r} \notin S \\ & \end{array}\right.\end{equation}$$(9) where |$S$| denotes support. It has been pointed out that the ER algorithm is closely related to the steepest descent method [6]. The name of this algorithm originates from a steady and quick reduction of the error, which is defined by the amount of violations of the constraints. However, in exchange for the strong convergence feature, the ER algorithm tends to suffer from stagnation at the local minimum solutions. To compensate for this drawback, another algorithm, called the hybrid input–output (HIO) algorithm, was devised [5, 6], in which the real-space constraint of Eq. (9) is modified as $$\begin{equation}{\psi _{n + 1}}\left( {{{\boldsymbol r}}} \right) = \left\{ \begin{array}{ll} \psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right) & {\boldsymbol r} \in S \\ {\psi _n}\left( {{{\boldsymbol r}}} \right) - \beta \psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right) & {\boldsymbol r} \notin S \\ \end{array}\right.\end{equation}$$(10) The name of this algorithm originates in part from the second line of Eq. (10), where the next ‘input’ |${\psi _{n + 1}}$| to the nonlinear operator due to the reciprocal-space constraint (see Fig. 2) is determined by the previous input |${\psi _n}$| and the previous ‘output’ |$\psi _n^{^{\prime}}$| in each iteration step. Thus, Eq. (10) as a whole is a ‘hybrid’ of the first line in the ER algorithm in Eq. (9). It is known that a proper value for parameter β is near 1 [6] and is empirically set as 0.5–1.8 in most cases. Although there is no mathematically rigorous proof of the uniqueness of the resulting solution [19], it is known that the HIO algorithm has the ability to escape from local minima and high tolerance to noise included in the diffraction data [5]. In most cases, an appropriate combination of ER and HIO algorithms is effectively used for rapid convergence to the final solution. The manner of reconstruction mentioned earlier is based on WPOA, and therefore, is suitable for phase imaging of atomic layer materials such as carbon nanotubes (CNTs) and graphene sheets [7–9, 11, 20], as summarized in Fig. 4. Figure 4a and b shows the first atomic-resolution imaging performed by EDI [7, 8]. Using a 200-kV FEG TEM, an isolated double-wall CNT found by ordinary low-resolution TEM observations was illuminated by a nearly parallel beam with a diameter of ∼50 nm to obtain the diffraction pattern shown in Fig. 4a. The area around the direct spot was replaced by the FT of the low-resolution TEM image, which was used for setting a rectangular support region so as to include the CNT. In the reconstructed image (the left panel in Fig. 4b), the outer and inner walls (red and yellow arrows) and the Moiré pattern between them, similar to those of the model (the right panel), were observed. The carbon bond length of 0.142 nm was resolved, despite a spatial resolution of HRTEM imaging of 0.22 nm for the microscope used for the experiment. Fig. 4. Open in new tabDownload slide Phase imaging of CNTs as weak-phase objects. (a) (b) First atomic-resolution results achieved by EDI for a double-wall CNT [7]. (c) (d) Phase imaging of a multi-wall CNT reconstructed from a 10-kV electron diffraction pattern in an SEM-based instrument [9]. (e) and (f) EDI of the atomic structure of a single-wall CNT reconstructed from a 30-kV electron diffraction pattern [20]. The magnified phase image trimmed from (f) is compared with a structure model in (g). (a)–(b), and (e)–(g) are reproduced with permission from the AAAS and AIP Publishing, respectively. CNT, carbon nanotube; EDI, electron diffractive imaging; SEM, scanning electron microscopy. Fig. 4. Open in new tabDownload slide Phase imaging of CNTs as weak-phase objects. (a) (b) First atomic-resolution results achieved by EDI for a double-wall CNT [7]. (c) (d) Phase imaging of a multi-wall CNT reconstructed from a 10-kV electron diffraction pattern in an SEM-based instrument [9]. (e) and (f) EDI of the atomic structure of a single-wall CNT reconstructed from a 30-kV electron diffraction pattern [20]. The magnified phase image trimmed from (f) is compared with a structure model in (g). (a)–(b), and (e)–(g) are reproduced with permission from the AAAS and AIP Publishing, respectively. CNT, carbon nanotube; EDI, electron diffractive imaging; SEM, scanning electron microscopy. The high-resolution capability of EDI is important, in particular, for observations of carbon-related materials using low-energy beams, in which a radiation damage is effectively suppressed but the quality of HRTEM imaging is strongly limited by objective lens (OL) aberration. The diffraction pattern in Fig. 4a was formed in the diffraction mode of the TEM. In other words, the reconstructed result is directly influenced by the diffraction-forming lenses (some of the intermediate lenses and the projector lens) and might be also influenced indirectly by the OL and other post-specimen lenses. To eliminate such uncertainty, an instrument based on a cold-FEG SEM was specially developed for 10- to 30-kV EDI [9, 11, 20]. In the instrument, a diffraction pattern created by a beam with a diameter of 100–150 nm is recorded as a ‘pure’ Fraunhofer diffraction pattern without employing any post-specimen lens. As shown in Fig. 4d, the side walls of a multi-wall CNT were reconstructed from the 10-keV diffraction pattern (Fig. 4c) after the saturated intensity of the direct spot was removed [9]. The compatibility of SEM imaging and diffraction recording was utilized to set a rectangular support region of appropriate dimensions. A wall separation of 0.34 nm, which corresponds to 30λ, was reconstructed based on the diffraction pattern, which was recorded up to 60 mrad. From a 30-keV diffraction pattern (Fig. 4e) obtained using the same instrument as that shown in Fig. 4c and d, the atomic structure of a single-wall CNT, as shown in Fig. 4f, was successfully reconstructed [20]. In Fig. 4g, the area trimmed from Fig. 4f shows good agreement with a structural model based on the chirality of (30, 16) determined from the diffraction pattern. The pairs of bright dots indicated by blue and magenta arrows have different intensities, which correspond to the isolated atoms and the atoms overlapping between the upper and lower walls, respectively. More precise and quantitative analyses may require a treatment beyond WPOA, because nanotubes exhibit a 3D morphology rather than a simple 2D shape. It is known that the WF that is weakly phase-modulated by the upper wall is also amplitude-modified owing to Fresnel propagation to the plane of the lower wall [21]. The quality of the result shown in Fig. 4f is substantially improved compared to that shown in Fig. 4d despite the work being conducted by the same researchers using the same instrument. It is generally known that the noise included in the diffraction data reduces the quality of the reconstructed result. As theoretically studied [22], WFs reconstructed from noisy diffraction data combined with random initial phases tend to be distributed in a spherical shell region around the correct solution in the 2N2-dimensional space. Therefore, averaging the reconstructed results started from many different initial-phase results in a noise-reduced WF quite close to the correct solution. The high-quality result shown in Fig. 4f is the result of such an averaging procedure. Reconstructions of complex wave fields Except for atomic-layer materials, such as CNTs and graphene sheets, WPOA is not valid for most of TEM samples. Therefore, the ability of reconstructing complex-valued WFs is highly required to increase the versatility of EDI. However, because of stagnations at the local minima, the reconstruction of a complex-valued object using the HIO algorithm is considerably more difficult than that of a real-valued non-negative object [23]. Under the non-negativity constraint, the pixel values in real space are assumed to be located on the positive half line of the real axis. However, in case of a complex WF, the pixel values are scattered over the 2D Argand plane, making it difficult to find the solution. Therefore, various real-space constraints in place of the non-negativity constraint and modification of the HIO algorithm have been devised for complex WF reconstruction. Fig. 5. Open in new tabDownload slide Reconstructions of the shapes of apertures and slits and the WFs therein. (a) TEM image, (b) diffraction pattern, and (c) reconstruction result of the pair of holes shown in (a) [24]. (d) SEM image of polygonal slits. (e) Bessel-like beam as the diffraction pattern and (f) the reconstructed phase distribution of the beam [25] with permission from the Oxford University Press on behalf of the Japanese Society of Microscopy. WFs, wave fields; TEM, transmission electron microscopy; SEM, scanning electron microscopy. Fig. 5. Open in new tabDownload slide Reconstructions of the shapes of apertures and slits and the WFs therein. (a) TEM image, (b) diffraction pattern, and (c) reconstruction result of the pair of holes shown in (a) [24]. (d) SEM image of polygonal slits. (e) Bessel-like beam as the diffraction pattern and (f) the reconstructed phase distribution of the beam [25] with permission from the Oxford University Press on behalf of the Japanese Society of Microscopy. WFs, wave fields; TEM, transmission electron microscopy; SEM, scanning electron microscopy. The first step was reconstructing the shape of the diffracting object. It has been clarified through simulation studies [23, 24] that an object having a simple symmetric shape, such as a circle, ellipse, or rectangle, cannot be reconstructed by the HIO algorithm, and that a complicated and asymmetric shape, such as a pair of separated areas, a doughnut shape with an off-centered hole, or a triangle, is desirable for a successful reconstruction. Based on these suggestions, the reconstruction of an aperiodic object by EDI was achieved for the first time in 2002 [24], 30 years after the phase retrieval of the periodic structure shown in Fig. 3. Figure 5a shows the TEM image of a pinhole pair fabricated by lithography, which has a separation of 100 nm and diameter of ∼30 nm in the sample plane. The diffraction pattern (Fig. 5b) from the pair shows interference fringes between the waves that passed through each hole. Based on the diffraction pattern, the shape of the diffracting objects (the hole pair) was successfully reconstructed, as shown in Fig. 5c. For the complex WF reconstruction, a support region larger than the objects and the constraint that the amplitude of the WF is confined to a constant value are used instead of the non-negativity constraint. However, the amplitude inside the holes is not uniform, unlike the TEM image shown in Fig. 5a. Such errors and the reduced resolution could have originated from the low signal-to-noise ratio (SNR) and beam coherence. In recent years, WF reconstructions inside apertures with various separated configurations have been applied to phase mapping in structured electron beams, such as vortex and Bessel beams [25]. Figure 5d shows a TEM image of polygonal slits fabricated by focused ion beams (FIBs). The WF inside the slits may have a slight phase modulation unless a parallel illumination. Based on the Bessel-like beam intensity shown in Fig. 5e, which is the diffraction pattern obtained from the slits, the phase distribution inside the slits was determined and Fourier-transformed to determine the phase distribution of the Bessel-like beam, as shown in Fig. 5f [25]. The diffractive imaging of pure phase objects, such as the interior of apertures and slits, may not be difficult as far as the diffraction data have been recorded with sufficiently high SNR and coherent illumination. However, the reconstruction of general complex WFs with amplitude and phase modulations requires an additional condition that the support shape should closely match the object shape [23, 24]. This is called ‘tight support’, to distinguish it from the ‘loose support’ explained previously, which includes both the object and a part of the surrounding constraint area. In CXDI, tight support is realized using ‘dynamic support’ techniques, where the initial loose support is dynamically updated during the iterative process to gradually converge to the tight support. A few algorithms for dynamic support have been proposed [10, 26, 27]. The importance of complicated and asymmetric shapes for diffracting objects was mentioned previously. Considering that most nanoparticles have simple shapes, such as a nearly spherical shape, not only using tight support but also removing the shape restriction are indispensable to the versatile use of EDI. As solutions, improvements in the phase retrieval algorithm, appropriate setting of the initial phase in the iteration, and addition of stronger constraints in the real space have been proposed, as explained in order below. Among some successful results obtained using the dynamic support [12, 17, 27, 28], the reconstruction of a complex WF from a symmetric particle based on the improved algorithm is explained by referring to Fig. 6a-c [12]. As shown in Fig. 6a, a cubic MgO smoke particle with a size of 20 nm supported on a carbon thin film was illuminated by an electron beam with a diameter of 150 nm to obtain a diffraction pattern from a direction near <100>. Strictly speaking, the target object must be isolated ideally in vacuum for the real-space constraint. Instead of such an unrealistic condition, the compact support is approximately realized by subtracting the scattering intensity from the supporting film, which is obtained from a neighboring area. The reconstruction was performed based on a combination of the dynamic support and an extended HIO algorithm, called guided HIO, where the best solution starting from multiple random initial phases is repeatedly used as inputs to the next-generation calculations [12]. Figure 6b shows the reconstructed result in which an array of atomic columns with a separation of 0.21 nm is clearly resolved. As observed in the magnified amplitude and phase images of Fig. 6c, the result shows good agreement with the multislice simulations embedded in the centers of each image. Fig. 6. Open in new tabDownload slide Reconstructions of complex WFs based on nanobeam diffraction patterns of crystalline particles. (a) An MgO particle illuminated by an electron beam with a diameter of 150 nm. (b) The reconstruction result and (c) detailed comparison with multislice simulations of the exit WF [12]. (d) Reconstruction of the crystalline structure of a TiO2 particle [13]. (e) Reconstructed result (left) and HRTEM image (right) of a CdS particle [14]. (a)–(c) and (d)–(e) are reproduced with permission from the AIP Publishing and Springer Nature (https://www.nature.com/nnano/andhttps://www.nature.com/nphys/), respectively. WFs, wave fields; HRTEM, high-resolution transmission electron microscopy. Fig. 6. Open in new tabDownload slide Reconstructions of complex WFs based on nanobeam diffraction patterns of crystalline particles. (a) An MgO particle illuminated by an electron beam with a diameter of 150 nm. (b) The reconstruction result and (c) detailed comparison with multislice simulations of the exit WF [12]. (d) Reconstruction of the crystalline structure of a TiO2 particle [13]. (e) Reconstructed result (left) and HRTEM image (right) of a CdS particle [14]. (a)–(c) and (d)–(e) are reproduced with permission from the AIP Publishing and Springer Nature (https://www.nature.com/nnano/andhttps://www.nature.com/nphys/), respectively. WFs, wave fields; HRTEM, high-resolution transmission electron microscopy. Compared to CXDI and EDI in an SEM (Figs. 4d, f, and g), the advantage of EDI in a TEM is that a magnified image of the object is readily available with high spatial resolution and can be utilized for the following three purposes. A general solution to stagnation problems is setting a good starting point for the iterative optimization. Therefore, instead of using a specialized algorithm like a guided HIO, a phase-contrast HRTEM image under the Scherzer condition has been effectively used as the initial phase map in the iteration for reconstructions of almost symmetric complex WFs [13–15, 27]. In addition, a TEM image is useful for setting a tight support shape without dynamic support calculations [13–15, 27]. In most cases, the edge of a particle can be sharply determined by tracing it or by intensity binarization of the TEM image, except for steep edges that exhibit a sort of edge contrast that appears under any focusing condition, as seen in Fig. 6a. Moreover, the lost information around the direct spot is recovered to some extent by placing the central area of the FT of the TEM image [7, 13–15]. Figure 6d and e shows examples in which these merits are utilized for reconstructing complex WFs transmitted through nanoparticles. The diffraction patterns obtained from a TiO2 particle (Fig. 6d) [13] and a CdS particle on an amorphous film or a graphene sheet (Fig. 6e) [14] were obtained using electron beams with diameters of ∼40 nm. In both cases, the scattering intensity obtained from a neighboring area was carefully subtracted as the background. The low-frequency information obtained by the FT of the TEM images was combined to the diffraction patterns. The TEM images were also used for setting the tight support, as described before. The remarkable feature in both reconstruction results is that the spatial resolution has been improved over those of the HRTEM images used as the initial images. In Fig. 6d, the reconstructed phase image shows the positions of oxygen columns (blue circles), which are not identified in the HRTEM image. Although the separation between Ti and O columns is difficult to recognize, the authors indirectly claim a 70 pm resolution based on the comparison between the model’s projected potential and (possibly) the averaged profile as shown in the right panel. Further quantitative discussion may require a precise comparison with simulations of the exit WF considering dynamical diffraction effects, as shown in Fig. 6c (and also later shown in Fig. 7) [12, 17, 28], because the phase image at the atomic level does not necessarily reflect the projected potential map beyond WPOA. On the other hand, the atomic columns of Cd and S with a separation of 84 pm have been clearly resolved in the reconstruction result shown in Fig. 6e. Reconstruction using a selector aperture As far as based on the real-space constraint that no materials should exist in the surrounding area, phase imaging by EDI is limited to isolated nanomaterials, such as nanotubes and nanoparticles supported on an adequately thin film. Considering that most targets of TEM observations are nanostructures embedded in materials such as interfaces and lattice defects, this restriction substantially suppresses the versatile use of the technique. Two solutions to this problem have been devised. The first is to employ a SAD instead of nanobeam diffraction, and the second is to employ ptychographic EDI. The former is described in this section and further in the following section, while the latter is described in the ‘Ptychographic EDI’ section. When using a SAD pattern for EDI, the dark area around the hole of the selector aperture can be utilized for the support. In this case, the constraint for the WF outside the support changes from ‘the real part of 1 and imaginary part of 0’ described in the ‘Basic principle’ section, to ‘both the real and imaginary parts of 0’. As this constraint and the tight support condition are always valid irrespective of the region selected by the aperture, reconstruction of not only isolated nanomaterials but any area of interest of a thin TEM sample becomes possible. As is evident in general TEM observations, the sampling interval in the sample plane due to a detector’s single pixel becomes finer when the sample is highly magnified, that is, a trade-off must always be created between the FOV and sampling interval. Because of this issue, as well as to achieve a large oversampling ratio, a selected area with a small diameter of <10 nm is favorable for precise atomic-scale reconstruction. Considering a typical OL magnification in TEMs, a pinhole with a diameter of several hundreds of nanometers fabricated in a metallic film by FIB acts as an ultrasmall selector aperture, as shown in Fig. 1c [28]. Because of the area-selection error induced by the geometric aberrations of the OL, an effective use of the ultrasmall aperture requires assistance by an imaging aberration corrector. This combined use of an ultrasmall aperture and an aberration corrector enables obtaining a SAD pattern from a well-defined nano area, which is referred to as ‘selected-area nanodiffraction (SAND)’ [29]. Besides the unrestricted sample shape, another advantage of SAD-based EDI is that it can easily achieve a high degree of spatial coherence of the illumination. When using a nanobeam for EDI, Even if the sample has an isolated shape, the beam diameter must be carefully adjusted to obtain the diffraction pattern. As seen in Fig. 6a, the beam should be small enough to avoid illuminating anything other than the sample, yet large enough to achieve a high degree of spatial coherence. It is known that the coherence length is approximately proportional to the beam diameter [30–32] and should be at least two times larger than the sample size for diffractive imaging [18], as mentioned earlier. When using a selector aperture instead of a nanobeam, almost completely coherent illumination can be achieved inside the aperture by enlarging the beam diameter such that it is much larger than the aperture. The degree of coherence inside a selector aperture can be estimated quantitatively by measuring the spatial coherence length determined by analyzing the Airy diffraction pattern of the aperture [31, 32]. Figure 7a shows a Si [011] SAND pattern obtained from an area near the edge of a wedge-shaped fragment. The area is thin enough for kinematical diffraction, as confirmed by the negligible intensity of the 200 forbidden spots. Based on a previous report [31], the beam diameter used to obtain the SAND pattern was enlarged to exceed 500 nm to achieve almost completely coherent illumination inside the ultrasmall aperture. Unlike the examples shown in Fig. 6, the reconstruction was conducted without using a TEM image, for the following reasons. Unlike the measurements shown in Figs. 4 and 6, there is no saturation of the direct spot intensity because the selected area is filled with the material. The saturation in Figs. 4 and 6 is induced because most of the illuminated regions are vacuum surrounding a particle or nanotube. In Fig. 7a, not only the direct spot but also all diffraction spots are convoluted with the aperture’s Airy pattern. Thus, the aperture shape information is directly recorded in the diffraction pattern, instead of being overwritten by the FT of the image. As a result, tight support is achieved easily and automatically by dynamic support rather than by carefully tracing the support outline in a TEM image. A possible disadvantage of using an aperture is that it is difficult to use an HRTEM image inside the aperture as the initial phase for the iterative calculation. This difficulty arises because a slight defocusing and image shift with respect to the aperture occur due to a change in the stray magnetic fields associated with the switching from imaging mode to diffraction mode [33]. Fig. 7. Open in new tabDownload slide SAND-based EDI of atomic structures in crystalline materials. (a) SAND pattern from a Si crystal from the [110] direction. (b) Multislice simulations of the exit WFs for various thicknesses. (c) Argand plot of the exit WFs. (d) and (e) Amplitude and phase images of the reconstructed WFs, respectively [17]. (f) and (g) Aberration-corrected HRTEM images of a Si crystal from the [110] and [112] directions [17]. (h) Reconstruction of the Si crystal structure from the [112] direction. (i) and (j) Reconstruction of the MgO crystal structure from the [100] and [110] directions, respectively [17]. Copyright (2019) by The Japan Society of Applied Physics. SAND, selected-area nanodiffraction; EDI, electron diffractive imaging; WFs, wave fields; HRTEM, high-resolution transmission electron microscopy. Fig. 7. Open in new tabDownload slide SAND-based EDI of atomic structures in crystalline materials. (a) SAND pattern from a Si crystal from the [110] direction. (b) Multislice simulations of the exit WFs for various thicknesses. (c) Argand plot of the exit WFs. (d) and (e) Amplitude and phase images of the reconstructed WFs, respectively [17]. (f) and (g) Aberration-corrected HRTEM images of a Si crystal from the [110] and [112] directions [17]. (h) Reconstruction of the Si crystal structure from the [112] direction. (i) and (j) Reconstruction of the MgO crystal structure from the [100] and [110] directions, respectively [17]. Copyright (2019) by The Japan Society of Applied Physics. SAND, selected-area nanodiffraction; EDI, electron diffractive imaging; WFs, wave fields; HRTEM, high-resolution transmission electron microscopy. Figure 7b shows multislice simulations of the exit WFs dynamically scattered by Si crystals with thicknesses ranging from 1.5 to 12.3 nm. As far as the thickness is <10 nm, both the amplitude and phase images are consistent with the atomic positions. The complex values at each pixel in the simulated WFs are examined in an Argand plot (Fig. 7c), where the phase modulations at any position stay within the range of 0–|${\rm{\pi }}$| for thicknesses <10 nm. Thus, the ‘non-negative imaginary part constraint’ is effective as the real-space constraint for the inside region of the aperture, instead of using an HRTEM image as the initial phase. Although the reconstruction quality obtained from various random initial phases was relatively poor [28], averaging these results in the same manner as that shown in Fig. 4f and g substantially improves the quality, making it possible to clearly identify the atomic structures in both the amplitude and phase images (Fig. 7d and e). The results show good agreement with the simulated WF for the thickness range of 4–6 nm [17]. Figure 7g and h shows a comparison between an aberration-corrected TEM image and a reconstructed WF transmitted through a silicon crystal along the [112] direction [17]. In the latter, the dumbbell separation of 78 pm formed by the 444 Bragg spots is clearly resolved, indicating a resolution exceeding the information limit of ∼100 pm of the aberration-corrected HRTEM (Fig. 7g) used for the SAND-EDI experiment. Figure 7i and j shows the reconstructed WF transmitted through MgO thin films along the [001] and [011] directions, respectively. The central squared regions in the selected areas of ∼6 nm in diameter are magnified in the panels on the right. Figure 7j shows an array of atomic columns having intensities corresponding to the respective atomic numbers. As generally known, the highest spatial frequency included in an HRTEM image is restricted up to the information limit mainly by the chromatic aberration of OL. That is, there are some unimaged diffraction beams beyond the information limit, and thus, information about the fine structure included in the transmitted WF is not included in the HRTEM images. On the other hand, the achievable spatial resolution of EDI should, in principle, reach the spatial frequency at which the Ewald sphere diverges from the zeroth-order Laue zone (ZOLZ). As seen in many examples introduced so far, a spatial resolution exceeding the information limit is achievable in EDI. Besides the issue of the information limit, HRTEM imaging has the difficulty in interpreting the actual atomic structures. Even if the sample is thin enough for kinematical diffraction, an artificial image contrast is often induced by an improper Fourier synthesis due to the oscillating phase-contrast transfer function (PCTF). In an aberration-corrected TEM, this problem is reduced to some extent by the suppression of the oscillation, but still remains as far as the PCTF is not flat, as clearly shown in Fig. 7f [34]. Moreover, in exchange for improving the point-to-point resolution, the aberration-corrected PCTF has absolute values much smaller than 1 in most of the frequency ranges. This means that most of the Bragg diffraction information is reduced in the image-forming process of an aberration-corrected HRTEM, resulting in reduced image contrast. One of the main characteristics of EDI is that atomic-resolution imaging is not directly influenced by geometric/chromatic aberrations of the OL. All diffraction beams included in ZOLZ are composed appropriately without any reduction or modulation by the PCTF, resulting in improved spatial resolution and image contrast and suppression of the artificial image contrast. In addition, by averaging the multiple solutions starting from various random initial phases, the SNR of the final image can be substantially improved, as mentioned earlier. The observability of light atoms, such as O columns in MgO (Fig. 7j) and single C atoms of single-wall CNTs (Fig. 4), is a significant advantage over annular dark-field scanning TEM (STEM) imaging. From the viewpoint of suppressing radiation damage, low-energy electron beams are effective for observing soft materials comprising light elements. However, atomic-resolution TEM/STEM imaging with a beam energy of several tens of kilovolts is strongly restricted by geometric and chromatic aberrations. Thus, one of the promising characteristics of EDI is the low-voltage observation of radiation-sensitive materials at atomic resolution, as shown in Fig. 4f and g. Phase imaging from a pair of TEM image and SAD pattern: returning to two-intensity measurements Atomic-scale phase imaging combined with an ultrasmall aperture has been extended to wide-FOV phase imaging using a conventional selector aperture with an effective diameter >100 nm [3, 17]. The information required for atomic-scale imaging is, typically, atomic positions with high resolution and ‘qualitative’ phase values for rough estimation of atomic numbers. On the other hand, the information required for wide-FOV phase imaging is ‘quantitative’ phase values reflecting continuous distributions of physical quantities. Following are the obstacles to achieving quantitative phase imaging across an FOV exceeding 100 nm: (i) need for precise measurements of small-angle scattering, (ii) increased influence of partial spatial coherence and lens aberration, and (iii) increase in inelastic scattering and the amount of phase modulation with increasing material thickness in the FOV. The details of these factors and the respective solutions are described below. As structural information on the length scale of ≥1–100 nm has low spatial frequencies, the small-angle scattering profile, which is generally recognized as a direct spot, must be precisely recorded with a sufficiently fine sampling interval. However, considering a typical detector pixel size, the maximum camera length in a conventional 200-kV TEM such as 200 cm is insufficient for such fine sampling. One of the solutions is to utilize additional magnification in the post-column energy filter, in which the diffraction pattern on the screen is magnified ∼8–10 times (Fig. 1d) [3, 32]. In exchange for such magnification, the Bragg spots go out of the detection angular region, resulting in loss of lattice information. This is another manifestation of the trade-off mentioned earlier. Thus, a medium-resolution phase image across an FOV of ∼100 nm is reconstructed from the small-angle scattering pattern. As with the original usage, the energy filter is used also to eliminate inelastic scattering from the sample. In an FOV of 100 nm, most TEM samples include areas thicker than several tens of nanometers, where non-negligible amounts of inelastic scattering are generated. It is thought that inelastic scattering reflects the absorption potential and is reduced in coherence. Therefore, zero-loss filtering is essential for phase imaging reflecting the elastic potential. It has been confirmed that the energy window of ±5 eV is sufficient for quantitative phase imaging [3, 17]. Another difficulty in EDI for thick materials is large phase modulations, which increase in proportion to the increase in thickness. Because the phase shift surpasses |${\rm{\pi }}$| in most cases, the non-negative imaginary part constraint used for atomic-resolution reconstruction is no longer valid. A promising choice for an alternative is using the TEM image intensity inside the selector aperture as a real-space constraint. Considering a small angle region is used for the reconstruction, the corresponding image should be a bright-field TEM (BF-TEM) image obtained with a small objective aperture and zero-loss filtering. As mentioned previously, the obstacles to using the HRTEM image intensity are a slight defocusing and slight image shift induced by switching from imaging mode to diffraction mode. Fortunately, a BF-TEM image is relatively insensitive to the defocusing. In addition, the image shift is relatively small compared to the aperture diameter of 100 nm and can be measured/compensated by using image shift coils if necessary [17]. Thus, unlike atomic-resolution imaging, the TEM image intensity inside the aperture can be used as a real-space constraint in medium-resolution phase imaging of wide FOVs (Fig. 1 d). For the ‘image-intensity constraint’, the ER and HIO algorithms of Eqs. (9) and (10) have been extended as $$\begin{equation}{\psi _{n + 1}}\left( {{{\boldsymbol r}}} \right) = \left\{ \begin{array}{ll} \sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} {{\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \over {\left| {\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \right|}} & {\boldsymbol r} \in A \\ 0 & {\boldsymbol r} \notin A \\ \end{array}\right.\end{equation}$$(11) and $$\begin{equation}{\psi _{n + 1}}\left( {{{\boldsymbol r}}} \right) \!=\! \left\{\!\!\!\! \begin{array}{ll} \left( {\left| {{\psi _n}\left( {{{\boldsymbol r}}} \right)} \right| - \beta \left\{ {\left| {\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \right| - \sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} } \right\}} \right) {{\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \over {\left| {\psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right)} \right|}} & {\boldsymbol r} \in A \\ {\psi _n}\left( {{{\boldsymbol r}}} \right) - \beta \psi _n^{^{\prime}}\left( {{{\boldsymbol r}}} \right) & {\boldsymbol r} \notin A \\ \end{array}\right.\end{equation}$$(12) respectively. Here, A signifies the area inside the aperture. As the entire |${\rm{N}} \times {\rm{N}}$| region is used as a real-space constraint, the definition of a support is no longer ambiguous. Therefore, in the extended algorithms, different equations are imposed on the regions inside and outside the aperture. The second lines of Eqs. (11) and (12) are the same as those of the ordinary ER and HIO algorithms, which make |$\psi \left( {{{\boldsymbol r}}} \right)$| outside the aperture converge to zero. In the first lines of Eqs. (9) and (10), the nth estimate |$\psi _n^{\rm{^{\prime}}}\left( {{{\boldsymbol r}}} \right)$| inside the support is used as the next input |${\psi _{n + 1}}\left( {{{\boldsymbol r}}} \right)$| without modification. In contrast, in Eqs. (11) and (12), only the phase of |$\psi _n^{\rm{^{\prime}}}\left( {{{\boldsymbol r}}} \right)$| is kept unmodified. The amplitude inside the aperture is replaced by |$\sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} $| in Eq. (11) and modified to converge to |$\sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} $| in Eq. (12) based on an expression derived by analogy to the second line of Eq. (10). Regardless of the amplitude convergence, the phase is continuously optimized to satisfy the reciprocal-space constraint. As a result, the input |${\psi _n}\left( {{{\boldsymbol r}}} \right)$| and output |$\psi _n^{\rm{^{\prime}}}\left( {{{\boldsymbol r}}} \right)$| converge to zero and |$\sqrt {{I^{image}}\left( {{{\boldsymbol r}}} \right)} $| outside and inside the aperture, respectively. It is well known that inserting a large objective aperture into the diffraction plane (the back focal plane of the OL) increases blurring of the HRTEM image because of the aberration of the image-forming lenses (mainly the OL). Similarly, inserting a large selector aperture into the image plane of the OL increases blurring of the diffraction pattern because of aberration of the diffraction-forming lenses located between the electron source and detector: the condenser lens, intermediate lens, projector lens, and lenses in the energy filter. To be precise, the sum of geometric aberrations of these lenses induces a complex-valued point-spread function (PSF) for the reciprocal-space WF. The resultant blurred diffraction pattern is further blurred by the real-valued PSF induced by partial spatial coherence, the detector, and other instrument instabilities [32]. As such blurring induces inconsistencies in the FT relation, the PSFs must be appropriately processed for quantitative phase reconstruction. These two types of PSFs can be precisely determined through profile analysis of the Airy pattern from a circular selector aperture inserted in a vacuum area [32]. As shown in Fig 8a-d, owing to the fitting calculations for the 2D profile of the blurred Airy pattern, the geometric aberration and width of the real-valued PSF are precisely determined. Generally, a deconvolution process tends to induce artifacts in the result. Fortunately, it has been confirmed that the blurred diffraction pattern can be appropriately deconvoluted by the determined real-valued PSF, unless there is too much blurring, that is, unless the spatial coherence length is smaller than the auto-correlation function of the aperture. Considering the convolution theorem, the real-space WF, which is reconstructed from the diffraction pattern blurred by the complex-valued PSF, should be the product of the transmitted WF for parallel illumination and the lens-transfer function. Therefore, the phase image of the sample is derived when the phase modulation calculated by the determined geometric aberration (Fig. 8d) is subtracted from the reconstructed phase image. After this processing, quantitative phase imaging with a phase accuracy of 0.1–0.2 rad is achieved as shown in Fig. 8e-h. Fig. 8. Open in new tabDownload slide Processing required for quantitative phase reconstruction. (a) Measured Airy pattern from a circular selector aperture and (b) fitting result to (a). The inset shows the real-valued PSF determined by the fitting calculation. (c) Comparisons among (a), (b), and the ideal Airy pattern. (d) Phase distribution due to geometric aberration determined by the fitting calculation [17, 32]. (e) TEM image of a crystal edge supported on a carbon film selected by an aperture. (f) Phase image reconstructed from the unprocessed SAD pattern. The arrows indicate artifacts. (g) Phase image after the quantification process. (h) Profile along the line in (g) [17]. Reproduced with permission from the Oxford University Press on behalf of the Japanese Society of Microscopy, and Copyright (2019) by The Japan Society of Applied Physics. PSF, point-spread function; TEM, transmission electron microscopy; SAD, selected-area diffraction. Fig. 8. Open in new tabDownload slide Processing required for quantitative phase reconstruction. (a) Measured Airy pattern from a circular selector aperture and (b) fitting result to (a). The inset shows the real-valued PSF determined by the fitting calculation. (c) Comparisons among (a), (b), and the ideal Airy pattern. (d) Phase distribution due to geometric aberration determined by the fitting calculation [17, 32]. (e) TEM image of a crystal edge supported on a carbon film selected by an aperture. (f) Phase image reconstructed from the unprocessed SAD pattern. The arrows indicate artifacts. (g) Phase image after the quantification process. (h) Profile along the line in (g) [17]. Reproduced with permission from the Oxford University Press on behalf of the Japanese Society of Microscopy, and Copyright (2019) by The Japan Society of Applied Physics. PSF, point-spread function; TEM, transmission electron microscopy; SAD, selected-area diffraction. Some examples of phase imaging of material thickness and electric fields around or inside materials are summarized in Fig. 9. Figure 9a and b shows a pair of amplitude and phase images of a wedge-shaped silicon crystal [3, 17]. The former is an energy-filtered BF-TEM image obtained under an off-Bragg condition, while the latter is reconstructed from a pair of the BF-TEM image and the energy-filtered SAD pattern shown in Fig. 9c. Reflecting the linear relation between the material thickness and phase shift of the transmitted beam, the phase gradient reconstructed in the wedge-shaped area shows good agreement with the thickness profile measured by electron energy loss spectroscopy, as shown in Fig. 9d. Figure 9e and f shows the observation of the electric field around MgO smoke particles [17]. As is often the case with insulator particles not in contact with a conductive supporting film, the MgO particle selected by the aperture is positively charged by the illumination electron beam. The phase gradients in the surrounding vacuum area reflect the electric fields radiated from the positively charged particles. Figure 9g-j shows the observation of a p–n junction in GaAs through visualization of the internal electrostatic potential [17]. In the phase image, the p–n junction with a band bending width of approximately 25 nm has been reconstructed. Considering the sample tilt from the zone axis to avoid strong dynamical diffraction effects, this value of the width agrees with the estimated value of 21 nm based on the depletion layer model for p–n junctions. Fig. 9. Open in new tabDownload slide Phase imaging from a pair of TEM image and SAD pattern. (a) The energy-filtered BF-TEM image and (b) phase image reconstructed from (a) and the energy-filtered SAD pattern in (c). (d) Comparison with the EELS measurement [3, 17]. (e) BF-TEM and (f) phase images showing the electric field around the MgO particles [17]. (g) BF-TEM image and (h) phase image reconstructed from (g) and the SAD pattern in (i). (j) Phase profile across the p–n junction in (h) [17]. Copyright (2019) by the Japan Society of Applied Physics. BF-TEM, bright-field transmission electron microscopy; SAD, selected-area diffraction; EELS, electron energy loss spectroscopy. Fig. 9. Open in new tabDownload slide Phase imaging from a pair of TEM image and SAD pattern. (a) The energy-filtered BF-TEM image and (b) phase image reconstructed from (a) and the energy-filtered SAD pattern in (c). (d) Comparison with the EELS measurement [3, 17]. (e) BF-TEM and (f) phase images showing the electric field around the MgO particles [17]. (g) BF-TEM image and (h) phase image reconstructed from (g) and the SAD pattern in (i). (j) Phase profile across the p–n junction in (h) [17]. Copyright (2019) by the Japan Society of Applied Physics. BF-TEM, bright-field transmission electron microscopy; SAD, selected-area diffraction; EELS, electron energy loss spectroscopy. As introduced in the ‘Phase imaging based on two-intensity measurements: Gerchberg–Saxton algorithm’ section, the image-intensity constraint was originally used in the GS algorithm. However, the scheme combined with the energy filter and improved algorithms is not merely a throwback to the two-intensity measurements in the dawn of the era of EDI. Compared to Fig. 3, the recently established EDI techniques have evolved so that not only periodic but also localized structures can be reconstructed, not only qualitatively but also quantitatively over an FOV exceeding 100 nm. Ptychographic EDI As a solution to reconstruct the phase image of a sample that is not surrounded by free space, reconstruction based on a SAD pattern was explained in the previous sections. In this section, ptychographic EDI is explained as an alternative solution [35]. In this method, multiple diffraction patterns are sequentially recorded from overlapping adjacent areas by moving a nanobeam probe or the sample. This is schematically shown in Fig. 1e, where only four areas are shown for clarity of explanation. In actual experiments, by increasing the number of areas, the FOV can be extended in exchange for increased measurement and calculation times. The exit WF below the sample formed by the illumination probe |$P\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right)$| located at the jth position |${{{{\boldsymbol R}}}_j}$| is described as $$\begin{equation}{\psi _j}\left( {{{\boldsymbol r}}} \right) = O\left( {{{\boldsymbol r}}} \right)P\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right),\end{equation}$$(13) where |$O\left( {{{\boldsymbol r}}} \right)$| is the object function, which is equivalent to the transmitted WF under parallel and uniform illumination. In this method, |$O\left( {{{\boldsymbol r}}} \right)$| consistent with the entire diffraction data set, |${I_j}\left( {{{\boldsymbol q}}} \right) = {\left| {\mathcal{F}\left[ {{\psi _j}\left( {{{\boldsymbol r}}} \right)} \right]} \right|^2}\left( {j = 1,2..} \right)$|⁠, is found through iterative calculations. In the reconstruction, instead of the HIO and ER algorithms, the extended ptychographic iterative engine (ePIE) expressed below [35] is used. $$\begin{equation}{O_{j + 1}}\left( {{{\boldsymbol r}}} \right) = {O_j}\left( {{{\boldsymbol r}}} \right) + {{{P^*}\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right)} \over {\left| {P\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right)} \right|_{max}^2}}\left\{ {\psi _j^{^{\prime}}\left( {{{\boldsymbol r}}} \right) - {\psi _j}\left( {{{\boldsymbol r}}} \right)} \right\}\end{equation}$$(14) Here, the change in the estimated WF from |${\psi _j}\left( {{{\boldsymbol r}}} \right)$| to |$\psi _j^{^{\prime}}\left( {{{\boldsymbol r}}} \right)$| is induced by the reciprocal space constraint based on the jth diffraction intensity, |${I_j}\left( {{{\boldsymbol q}}} \right)$|⁠. Considering Eq. (13), the corresponding change in the object function is given by |$\left\{ {\psi _j^{^{\prime}}\left( {{{\boldsymbol r}}} \right) - {\psi _j}\left( {{{\boldsymbol r}}} \right)} \right\}/P\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right)$|⁠. In addition, the term is weighted by the normalized probe intensity |${\left| {P\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right)} \right|^2}/\left| {P\left( {{{{\boldsymbol r}}} - {{{{\boldsymbol R}}}_j}} \right)} \right|_{max}^2$|to avoid it being influenced by the noise in the low-intensity regions of the probe. As a result, the update of the object function |${O_j}\left( {{{\boldsymbol r}}} \right)$| based on the jth diffraction pattern is given by Eq. (14). By sequentially applying the same procedure to all diffraction patterns, the overall object function |$O\left( {{{\boldsymbol r}}} \right)$| covering the entire measured area is updated. This update is iteratively repeated to find the final solution of |$O\left( {{{\boldsymbol r}}} \right)$|⁠, which has the least confliction with all diffraction data sets. Thus, the overlapping areas between the adjacent probe areas are used as a type of real-space constraint in ptychograpic EDI. Another merit of the method is that the complex-valued probe function, which is generally difficult to determine, can be optimized by similar iterative calculations using the equation obtained by exchanging |$O$| and |$P$| in Eq. (14). The reconstruction quality is largely improved by this optimization [35]. Successful examples of ptychographic EDI are summarized in Fig. 10. Figure 10a and b is probably the first result in which the phase image of FeNi particles several tens of nanometers in size have been reconstructed across an FOV of ∼90 nm [36]. To obtain the diffraction data set, an electron probe with a diameter of 40 nm was moved along the |$6 \times 6$| grid positions, maintaining an overlap with the neighboring probe positions, as shown in Fig. 10a (only four positions are highlighted). Instead of moving the illumination probe, moving the sample is also feasible. Figure 10c and d shows phase imaging of polystyrene spheres, 260 nm in diameter, attached to a carbon film [37]. The sample holder was translated to each of the 400 grid points to shift the relative position between the sample and selector aperture, which has an effective diameter of ∼200 nm (schematically shown at the bottom left of Fig. 10c). Based on the set of SAD patterns, a phase image in which the maximum deviation exceeds 6|${\rm{\pi }}$| is reconstructed across the 1-|${\rm{\mu m}}$| FOV (Fig. 10d). In the same manner as the two preceding experiments conducted in conventional 200-kV FEG TEMs, ptychographic EDI has also been achieved in a 30-kV FEG SEM specially equipped with a detector at the downstream of the sample plane [38]. Figure 10e shows a phase image of Au particles supported by a carbon film, which was reconstructed from the diffraction data set obtained from |$20 \times 20$| grid points scanned by an SEM probe 15 nm in diameter. In a subsequent study, the diffraction data was additionally processed to eliminate the blurring originating from partial coherence in the probe and the detector PSF [39]. As shown in Fig. 10f, clear lattice fringes have been finally reconstructed, indicating a spatial resolution superior to that of the original SEM imaging. There are only a few reports in which atomic resolution has been achieved by the ptychographic EDI [40–42]. Figure 10g shows the state-of-the-art result of the atomic-resolution ptychographic EDI of twisted bilayer MoS2 reconstructed using an 80-kV aberration-corrected STEM probe scanning along a |$68 \times 68$| grid at the interval of 59 pm. In the phase image, atomic spacings of 61 and 42 pm are recognized as a separated peak pair and by a dip in the profile, respectively. The important point for the significantly high resolution is to record up to a high scattering angle where the intensity is almost disappeared. For the experiment of Fig. 10g, such ‘full-field ptychography’ was achieved using a high-speed pixel-array detector specially developed for high detective quantum efficiency (DQE) [42]. Fig. 10. Open in new tabDownload slide Phase imaging by ptychographic EDI. (a) ADF-STEM image and (b) phase image of FeNi particles [36]. (c) TEM image and (d) phase image of polystyrene spheres [37]. (e) Phase image of Au particles on a carbon film reconstructed using an SEM probe [38]. (f) The improved result from (e) by additional processing to remove the diffraction blurring [39]. (g) Atomic-resolution reconstruction of twisted bilayer MoS2 [42]. (a)–(b), (c)–(d), and (g) are reproduced with the permission of Copyright (2007) by the American Physical Society (https://doi.org/10.1103/physrevb.82.121415), under the Creative Commons licenses, and from Springer Nature (https://www.nature.com/), respectively. EDI, electron diffractive imaging; ADF-STEM, annular dark-field scanning TEM; TEM, transmission electron microscopy; SEM, scanning electron microscopy. Fig. 10. Open in new tabDownload slide Phase imaging by ptychographic EDI. (a) ADF-STEM image and (b) phase image of FeNi particles [36]. (c) TEM image and (d) phase image of polystyrene spheres [37]. (e) Phase image of Au particles on a carbon film reconstructed using an SEM probe [38]. (f) The improved result from (e) by additional processing to remove the diffraction blurring [39]. (g) Atomic-resolution reconstruction of twisted bilayer MoS2 [42]. (a)–(b), (c)–(d), and (g) are reproduced with the permission of Copyright (2007) by the American Physical Society (https://doi.org/10.1103/physrevb.82.121415), under the Creative Commons licenses, and from Springer Nature (https://www.nature.com/), respectively. EDI, electron diffractive imaging; ADF-STEM, annular dark-field scanning TEM; TEM, transmission electron microscopy; SEM, scanning electron microscopy. Concluding remarks Inheriting the basic idea by Gerchberg and Saxton, various experimental processes with correspondingly appropriate constraints have been devised to advance the EDI. Now that we have entered the stage of application to advanced measurements, it is important to appropriately apply these processes and constraints according to the target samples and types of information to be measured so that each characteristic can be fully utilized. High-resolution phase imaging, unaffected by chromatic aberration, is promising, especially for examining radiation-sensitive materials at low acceleration voltages. Processes based on single-intensity measurements are advantageous in terms of their temporal resolution. In the future, application to pulse electron beams, as is currently being worked on with XFELs, may be possible. On the other hand, processes based on two/multi-intensity measurements, such as the SAD-based EDI and ptychographic EDI, are preferable for quantitative phase imaging and available for most TEM samples. The applicability to extended materials is advantageous over off-axis electron holography, where a reference vacuum area is required near the area of interest. Another merit of EDI is that, except for some examples introduced in this article, experiments for thin samples can be conducted basically in the current analytical electron microscopes equipped with the FEG. 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TI - Wave field reconstruction and phase imaging by electron diffractive imaging
JF - Microscopy
DO - 10.1093/jmicro/dfaa063
DA - 2021-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/wave-field-reconstruction-and-phase-imaging-by-electron-diffractive-e0VO0oyHBU
SP - 116
EP - 130
VL - 70
IS - 1
DP - DeepDyve
ER -