Exit wave reconstruction at atomic resolution

An iterative method for exit wave function reconstruction based on wave function propagation in free space is presented. The method, which has the potential for application to many forms of microscopy, has been tailored to work with a through focal series of images measured in a high-resolution transmission electron microscope. Practical difficulties for exit wave reconstruction which are pertinent in this experimental environment are the slight incoherence of the electron beam, sample drift and its effect upon the defocus step size that can be utilised, and the number of image measurements that need to be made. To gauge the effectiveness of the method it is applied to experimental data that has been analysed previously using a maximum likelihood formalism (the MAL method).     

Correction of autofocusing errors due to specimen tilt for automated electron tomography

Transmission electron microscopy images acquired under tilted‐beam conditions experience an image shift as a function of defocus settings – a fact that is exploited as a method for defocus determination in most of the automated tomography data collection systems. Although the method was shown to be highly accurate for a large variety of specimens, we point out that in its original design it can strictly only be applied to images of untilted samples. The application to tilted samples and thus in automated electron tomography is impaired mainly due to a defocus change across the images, resulting in reduced accuracy. In this communication we present a method that can be used to improve the accuracy of the basic autofocusing procedures currently used in systems for automated electron tomography.     

A new method for the determination of the wave aberration function for high-resolution TEM.; 2. Measurement of the antisymmetric aberrations

A new method is presented for the determination of the antisymmetric coefficients of the wave aberration function from a tableau of tilted illumination images. The approach is based on measurements of the apparent defocus and two-fold astigmatism using a phase correlation function and phase contrast index calculated from a short focus series acquired at each tilt. This method is shown to be suitable for a wide range of specimens and is sufficiently accurate for exit plane wave restoration at 0.1 nm resolution. Experimental examples of this approach are provided and the method is compared to results obtained from measurements of conventional power spectra.     

A new method for the determination of the wave aberration function for high resolution TEM 1. Measurement of the symmetric aberrations

A new method for the accurate determination of the symmetric coefficients of the wave aberration function has been developed. The relative defoci and displacements of images in a focus series are determined from an analysis of the phase correlation function between pairs of images, allowing the restoration of an image wave even when focus and specimen drift are present. Subsequently, the absolute coefficients of both defocus and 2-fold astigmatism are determined with a phase contrast index function. Overall this method allows a very accurate automated aberration determination even for largely crystalline samples with little amorphous contamination. Using experimental images of the complex oxide Nb16W18O94 we have demonstrated the new method and critically compared it with existing diffractogram based aberration determinations. A series of protocols for practical implementation is also given together with a detailed analysis of the accuracy achieved. Finally a focal series restoration of Nb16W18O94 with symmetric aberrations determined automatically using this method is presented.     

A novel method for focus assessment in atomic resolution STEM HAADF experiments

Analysis of the Fourier components of through-focal images in scanning transmission electron microscopy with a high angle annular dark field detector is used to assess illumination defocus values. The method is based on a least squares fitting of the peculiar dependence of Fourier components of the high angle annular dark field image on defocus. The validity of the method has been checked against simulations and experiments obtaining a good level of accuracy on the defocus measurement (δf=2 nm) for simulated specimen thickness up to 40 nm. The difference between simulated and experimental Fourier coefficients for large defoci can be used to estimate the specimen thickness at least up to 30 nm but with decreasing precision for larger thickness.     

Compositional analysis of mixed–cation-anion III–V semiconductor interfaces using phase retrieval high-resolution transmission electron microscopy

Employing exit‐plane wave function (EPWF) reconstruction in high‐resolution transmission electron microscopy (HRTEM), we have developed an approach to atomic scale compositional analysis of III‐V semiconductor interfaces, especially suitable for analyzing quaternary heterostructures with intermixing in both cation and anion sub‐lattices. Specifically, we use the focal‐series reconstruction technique, which retrieves the complex‐valued EPWF from a thru‐focus series of HRTEM images. A study of interfaces in Al_0.4Ga_0.6As–GaAs and In_0.25Ga_0.75Sb–InAs heterostructures using focal‐series reconstruction shows that change in chemical composition along individual atomic columns across an interface is discernible in the phase image of the reconstructed EPWF. To extract the interface composition profiles along the cation and anion sub‐lattices, quantitative analysis of the phase image is performed using factorial analysis of correspondence. This enabled independent quantification of changes in the In–Ga and As–Sb contents across ultra‐thin interfacial regions (approximately 0.6 nm wide) with true atomic resolution, in the In_0.25Ga_0.75Sb–InAs heterostructure. The validity of the method is demonstrated by analyzing simulated HRTEM images of an InAs–GaSb–InAs model structure with abrupt and graded interfaces. Our approach is general, permitting atomic‐level compositional analysis of heterostructures with two species per sub‐lattice, hitherto unfeasible with existing HRTEM methods.     

Object wave reconstruction by phase-plate transmission electron microscopy

A method is described for the reconstruction of the amplitude and phase of the object exit wave function by phase-plate transmission electron microscopy. The proposed method can be considered as in-line holography and requires three images, taken with different phase shifts between undiffracted and diffracted electrons induced by a suitable phase-shifting device. The proposed method is applicable for arbitrary object exit wave functions and non-linear image formation. Verification of the method is performed for examples of a simulated crystalline object wave function and a wave function acquired with off-axis holography. The impact of noise on the reconstruction of the wave function is investigated.     

A practical approach for STEM image simulation based on the FFT multislice method

It has been demonstrated that a high-angle annular dark-field (HAADF) STEM technique gives an image resolving atomic columns. Due to the diffusion of this technique and an improvement of its resolution, a practical procedure for image simulation becomes important for a quantitative interpretation of the HAADF image. In this report a new practical scheme for a STEM image simulation is developed based on the FFT multislice algorithm. Here, a HAADF intensity due to thermal diffuse scattering (TDS) is calculated from the absorptive potential corresponding to high-angle TDS and the wave function equivalent to the propagating probe within the sample. Contrary to the commonly used Bloch wave method, a coherent bright-field intensity and a coherent HAADF intensity are also obtained straightforwardly. The HAADF image contrast calculated for GaAs is not simply proportional to Z2 as expected from the Rutherford scattering at high-angle, and the As/Ga contrast ratio depends on the specimen thickness. This suggests that the generation of the HAADF signal is appreciably affected by the coherent dynamical scattering. The developed procedure here will have a definitive advantage over the Bloch wave approach for simulating the HAADF images expected from a defect and interface or amorphous materials, and also the HAADF image obtained by using a Cs-corrected microscope. This is because the former requires a huge super cell, while the latter needs a large objective aperture including a large number of incident beam directions.     

离焦模糊图像的维纳滤波复原研究

微操作中，显微视觉系统获取的图像通常是离焦模糊图像。离焦模糊图像的退化模型可用圆盘函数描述，利用模糊图像无方向性的二阶拉氏微分图像的自相关的负相关峰形成的环形槽的直径等于作为圆盘函数直径的2倍可以确定该函数。对模糊图像进行一次维纳滤波方法得到原图像的估计值，然后利用该初始值求得原图像及噪声的谱密度估值，进而利用这些新获得的信息构成改进的维纳滤波器对退化图像进行第二次滤波。实验表明，该方法计算量小、鉴别精度高、抗噪声能力较强，突出原图像的一些关键细节，提高了图像的复原质量。     

Bloch wave-based calculation of imaging properties of high-resolution scanning confocal electron microscopy

An efficient, Bloch wave-based method is presented for simulation of high-resolution scanning confocal electron microscopy (SCEM) images. The latter are predicted to have coherent nature, i.e. to exhibit atomic contrast reversals depending on the lens defocus settings and sample thickness. The optimal defocus settings are suggested and the 3D imaging capabilities of SCEM are analyzed in detail. In particular, by monitoring average image intensity as a function of the probe focus depth, it should be possible to accurately measure the depth of a heavy-atom layer embedded in a light-element matrix.     

Experimental quantification of annular dark-field images in scanning transmission electron microscopy

This paper reports on a method to obtain atomic resolution Z-contrast (high-angle annular dark-field) images with intensities normalized to the incident beam. The procedure bypasses the built-in signal processing hardware of the microscope to obtain the large dynamic range necessary for consecutive measurements of the incident beam and the intensities in the Z-contrast image. The method is also used to characterize the response of the annular dark-field detector output, including conditions that avoid saturation and result in a linear relationship between the electron flux reaching the detector and its output. We also characterize the uniformity of the detector response across its entire area and determine its size and shape, which are needed as input for image simulations. We present normalized intensity images of a SrTiO(3) single crystal as a function of thickness. Averaged, normalized atom column intensities and the background intensity are extracted from these images. The results from the approach developed here can be used for direct, quantitative comparisons with image simulations without any need for scaling.     

Imaging a dense nanodot assembly by phase retrieval from TEM images

Phase retrieval is a classical inverse problem in many fields dealing with waves that is becoming of increasing interest in transmission electron microscopy (TEM). A non-interferometric approach is here applied to TEM images. Phase retrieval possibilities given by the transport intensity equation are compared to the ones deriving from the weak phase object approximation. In the limit of small angles, both methods lead to a similar equation between the phase and a set of defocus images. This equation can be solved by an image processing equivalent to using a specific filter in Fourier space. This processing leads to phase images with a spatial resolution here essentially limited by the defocus amount between images. A dense assembly of silicon nanodots is used as a model case to illustrate the interest of this approximate phase retrieval method which can be carried out on standard equipment. The dot heights estimated using the phase images are found to be in good agreement with ones measured by atomic force microscopy. Since image noise and large defocus values may strongly affect the solution given by the approximate method, an iterative phase retrieval method is also used as a test for working conditions.     

Essential experimental parameters for quantitative structure analysis using spherical aberration-corrected HAADF-STEM

The accuracy of quantitative analysis for Z-contrast images with a spherical aberration (C_s) corrected high-angle annular dark-field (HAADF) scanning transmission electron microscope (STEM) using SrTiO₃(0 0 1) was systematically investigated. Atomic column and background intensities were measured accurately from the experimental HAADF-STEM images obtained under exact experimental condition. We examined atomic intensity ratio dependence on experimental conditions such as defocus, convergent semi-angles, specimen thicknesses and digitalized STEM image acquisition system: brightness and contrast. In order to carry out quantitative analysis of C_s-corrected HAADF-STEM, it is essential to determine defocus, to measure specimen thickness and to fix setting of brightness, contrast and probe current. To confirm the validity and accuracy of the experimental results, we compared experimental and HAADF-STEM calculations based on the Bloch wave method.     

Refinement procedure for the image alignment in high-resolution electron tomography

High-resolution electron tomography from a tilt series of transmission electron microscopy images requires an accurate image alignment procedure in order to maximise the resolution of the tomogram. This is the case in particular for ultra-high resolution where even very small misalignments between individual images can dramatically reduce the fidelity of the resultant reconstruction. A tomographic-reconstruction based and marker-free method is proposed, which uses an iterative optimisation of the tomogram resolution. The method utilises a search algorithm that maximises the contrast in tomogram sub-volumes. Unlike conventional cross-correlation analysis it provides the required correlation over a large tilt angle separation and guarantees a consistent alignment of images for the full range of object tilt angles. An assessment based on experimental reconstructions shows that the marker-free procedure is competitive to the reference of marker-based procedures at lower resolution and yields sub-pixel accuracy even for simulated high-resolution data.     

Contrast In Transmission Electron Micrographs Of Void Arrays In Crystals

In order to understand the detailed features in electron micrographs from radiation induced void lattices in crystals, an attempt has been made to determine the contrast from columns of regularly spaced strain-free spherical voids using calculations based on the two-beam approximation. Analytical expressions have been obtained for the wave amplitudes at the exit surface of a model crystal using matrix algebra. Line profiles for the contrast across a void column have been computed and two-dimensional images simulated on a line printer. The contrast behaviour has been analysed in terms of deviation from the Bragg condition for varying void column parameter: void distance/void radius. For dynamical, as well as for kinematical diffraction conditions the influence on the image of defocusing phase contrast has been investigated by addition of extra phase contributions to the diffracted beam amplitudes. Defocusing in most cases induces an enhanced contrast from the void column.     

Improved Zernike‐type phase contrast for transmission electron microscopy

Zernike phase contrast has been recognized as a means of recording high‐resolution images with high contrast using a transmission electron microscope. This imaging mode can be used to image typical phase objects such as unstained biological molecules or cryosections of biological tissue. According to the original proposal discussed in Danev and Nagayama (2001) and references therein, the Zernike phase plate applies a phase shift of π/2 to all scattered electron beams outside a given scattering angle and an image is recorded at Gaussian focus or slight underfocus (below Scherzer defocus). Alternatively, a phase shift of ‐π/2 is applied to the central beam using the Boersch phase plate. The resulting image will have an almost perfect contrast transfer function (close to 1) from a given lowest spatial frequency up to a maximum resolution determined by the wave length, the amount of defocus and the spherical aberration of the microscope. In this paper, I present theory and simulations showing that this maximum spatial frequency can be increased considerably without loss of contrast by using a Zernike or Boersch phase plate that leads to a phase shift between scattered and unscattered electrons of only π /4, and recording images at Scherzer defocus. The maximum resolution can be improved even more by imaging at extended Scherzer defocus, though at the cost of contrast loss at lower spatial frequencies.     

Direct exit-wave reconstruction from a single defocused image

We propose a direct, non-iterative method for the exact recovery of the complex wave in the exit-surface plane of a coherently illuminated object from a single defocused image. The method is applicable for a wide range of illumination conditions. The defocus range is subject to certain conditions, which if satisfied allow the complex exit-surface wave to be directly recovered by solving a set of linear equations. These linear equations, whose coefficients depend on the incident illumination, are obtained by analyzing the autocorrelation function of an auxiliary wave which is related to the exit-surface wave in a simple way. This autocorrelation is constructed by taking the inverse Fourier transform of the defocused image. We present an experimental proof of concept by recovering the exit-surface wave of a microfiber illuminated by a plane wave formed using a HeNe laser.     

Diagnosis of aberrations from crystalline samples in scanning transmission electron microscopy

The Ronchigrams, or shadow images, observed from a thin crystalline sample in a scanning transmission electron microscope characteristically present many sets of fringes, which appear thanks to the coherent interference between the various Bragg-diffracted discs as they overlap in the diffraction plane. A particular region of such patterns can be shown to be independent of the defocus at which they are recorded. The intensity along this so-called achromatic ring depends on the microscope aberrations and can be used to diagnose the wave aberration coefficients, a crucial first step in the operation of an aberration-corrected microscope. A new algorithm is presented that allows the accurate determination of all non-cylindrically symmetric aberrations up to fourth-order from a crystalline sample using this property of the Ronchigram. An experimental procedure for determining the position of and intensity along the achromatic lines, as well as examples of diagnosis from two different crystalline structures, are detailed.     

Diagnosis of aberrations from crystalline samples in scanning transmission electron microscopy

The Ronchigrams, or shadow images, observed from a thin crystalline sample in a scanning transmission electron microscope characteristically present many sets of fringes, which appear thanks to the coherent interference between the various Bragg-diffracted discs as they overlap in the diffraction plane. A particular region of such patterns can be shown to be independent of the defocus at which they are recorded. The intensity along this so-called achromatic ring depends on the microscope aberrations and can be used to diagnose the wave aberration coefficients, a crucial first step in the operation of an aberration-corrected microscope. A new algorithm is presented that allows the accurate determination of all non-cylindrically symmetric aberrations up to fourth-order from a crystalline sample using this property of the Ronchigram. An experimental procedure for determining the position of and intensity along the achromatic lines, as well as examples of diagnosis from two different crystalline structures, are detailed.