Projected potential profiles across interfaces obtained by reconstructing the exit face wave function from through focal series

An iterative method for reconstructing the exit face wave function from a through focal series of transmission electron microscopy image line profiles across an interface is presented. Apart from high-resolution images recorded with small changes in defocus, this method works also well for a large defocus range as used for Fresnel imaging. Using the phase-object approximation the projected electrostatic as well as the absorptive potential profiles across an interface are determined from this exit face wave function. A new experimental image alignment procedure was developed in order to align images with large relative defocus shift. The performance of this procedure is shown to be superior to other image alignment procedures existing in the literature. The reconstruction method is applied to both simulated and experimental images.     

Sub-Angstrom high-resolution transmission electron microscopy at 300 keV.

Sub-Angstrom transmission electron microscopy has been achieved at the National Center for Electron Microscopy (NCEM) by a one-Angstrom microscope (OAM) project using software and enhanced hardware developed within a Brite-Euram project (Ultramicroscopy 64 (1996) 1). The NCEM OAM provides materials scientists with transmission electron microscopy at a resolution better than 1 A by using extensive image reconstruction to exploit the significantly higher information limit of an FEG-TEM over its Scherzer resolution limit. Reconstruction methods chosen used off-axis holograms and focal series of underfocused images. Measured values of coherence parameters predict an information limit of 0.78 A. Images from a [1 1 0] diamond test specimen show that sub-Angstrom resolution of 0.89 A has been achieved with the OAM using focal series reconstruction.     

Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation

Through focus series of images are collected from MgO nano-cube crystals in the transmission electron microscope (TEM). The experimental data is used to solve the transport of intensity equation (TIE) to retrieve phase maps, which portray the morphology of the cubes and are quantified by the mean inner potential V(0). Particular attention is given to the practical difficulties associated with TIE phase retrieval of non-conducting polyhedron particles.     

Astigmatic intensity equation for electron microscopy based phase retrieval

Phase retrieval, in principle, can be performed in a transmission electron microscope (TEM) using arbitrary aberrations of electron waves; provided that the aberrations are well-characterised and known. For example, the transport of intensity equation (TIE) can be used to infer the phase from a through-focus series of images. In this work an "astigmatic intensity equation" (AIE) is considered, which relates phase gradients to intensity variations caused by TEM objective lens focus and astigmatism variations. Within the paraxial approximation, it is shown that an exact solution of the AIE for the phase can be obtained using efficient Fourier transform methods. Experimental requirements for using the AIE are the measurement of a through-focus derivative and another intensity derivative, which is taken with respect to objective lens astigmatism variation. Two quasi-experimental investigations are conducted to test the validity of the solution.     

Transmission electron microtomography without the "missing wedge" for quantitative structural analysis

A three-dimensional (3D) visualization and structural analysis of a rod-shaped specimen of a zirconia/polymer nanocomposite material were carried out by transmission electron microtomography (TEMT) with particular emphasis on complete rotation of the specimen (tilt angular range: +/-90 degrees ). In order to achieve such an ideal experimental condition for the TEMT, improvements in the specimen as well as the sample holder were made. A rod-shaped specimen was necessary in order to obtain a high transmission of the specimen upon tilting to large angles. The image resolution of the reconstructed tomogram was isotropic, in sharp contrast to the anisotropic image resolution of the conventional TEMT with a limited angular range (the "missing wedge" problem). A volume fraction of zirconia, phi, evaluated from the 3D reconstruction was in quantitative agreement with the known composition of the nanocomposite. A series of 3D reconstructions was made from the tilt series with complete rotation by limiting the maximum tilt angle, alpha, from which a couple of structural parameters, the volume fraction and surface area per unit volume, Sigma, of the zirconia, were evaluated as a function of alpha. It was confirmed from actual experimental data that both phi and Sigma slightly decreased with the increasing alpha and reached constant values at around alpha=80 degrees , suggesting that the specimen may have to be tilted to +/-80 degrees for truly quantitative measurements.     

Upgrading a microscope with a spiral phase plate

We present the implementation of a spiral phase plate in a standard bright-field microscope to enhance the contrast of phase and amplitude samples. The method can be employed in all types of microscopy where standard phase contrast methods are applicable, for example, in bright-field transmission or reflection microscopy using an illumination source with partial spatial coherence. The spiral phase filter is placed into an accessible Fourier plane of the imaging path of the microscope. It is shown that this produces not only a strong contrast enhancement but in theory also improves the spatial resolution of the microscope for white light. A series of different set-ups for transmissive or reflective samples, including epi-illumination, are presented to demonstrate the practical range of applications of this contrasting method. A minute shift of the spiral phase plate out of the centre results in relief-like images that are similar to those obtained by differential interference contrast microscopy. A series of such relief-like images can be numerically processed to obtain quantitative phase and amplitude information of the sample.     

Vector field electron tomography of magnetic materials: theoretical development

The theory of vector field electron tomography, the reconstruction of the three-dimensional magnetic induction around a magnetized object, is derived within the framework of Lorentz transmission electron microscopy. The tomographic reconstruction method uses as input two orthogonal tilt series of magnetic phase maps and is based on the vector slice theorem. An analytical reconstruction of the magnetic induction of a single magnetic dipole is presented as a proof-of-concept. The method is compared to two previously reported approaches: a reconstruction starting from the gradient of the magnetic phase maps, and a direct reconstruction of the magnetic vector potential. Numerical examples as well as estimates of the reconstruction errors for a range of magnetic particle shapes are reported.     

Off-axis and inline electron holography: A quantitative comparison

Capable of quantitatively imaging static magnetic and electric potentials and even strain electron holography is a very versatile and powerful TEM technique. In this paper we compare off-axis electron holography with a recently developed focal series reconstruction algorithm and phase retrieval based on the transport of intensity equation. Based on theoretical considerations and simulations we compare the different techniques with respect to parameters such as the coherence requirements, field of view, resolution, noise properties, and other required experimental conditions.     

Linear versus non-linear structural information limit in high-resolution transmission electron microscopy

A widely used performance criterion in high-resolution transmission electron microscopy (HRTEM) is the information limit. It corresponds to the inverse of the maximum spatial object frequency that is linearly transmitted with sufficient intensity from the exit plane of the object to the image plane and is limited due to partial temporal coherence. In practice, the information limit is often measured from a diffractogram or from Young's fringes assuming a weak phase object scattering beyond the inverse of the information limit. However, for an aberration corrected electron microscope, with an information limit in the sub-angstrom range, weak phase objects are no longer applicable since they do not scatter sufficiently in this range. Therefore, one relies on more strongly scattering objects such as crystals of heavy atoms observed along a low index zone axis. In that case, dynamical scattering becomes important such that the non-linear and linear interaction may be equally important. The non-linear interaction may then set the experimental cut-off frequency observed in a diffractogram. The goal of this paper is to quantify both the linear and the non-linear information transfer in terms of closed form analytical expressions. Whereas the cut-off frequency set by the linear transfer can be directly related with the attainable resolution, information from the non-linear transfer can only be extracted using quantitative, model-based methods. In contrast to the historic definition of the information limit depending on microscope parameters only, the expressions derived in this paper explicitly incorporate their dependence on the structure parameters as well. In order to emphasize this dependence and to distinguish from the usual information limit, the expressions derived for the inverse cut-off frequencies will be referred to as the linear and non-linear structural information limit. The present findings confirm the well-known result that partial temporal coherence has different effects on the transfer of the linear and non-linear terms, such that the non-linear imaging contributions are damped less than the linear imaging contributions at high spatial frequencies. This will be important when coherent aberrations such as spherical aberration and defocus are reduced.     

Measurement of effective source distribution and its importance for quantitative interpretation of STEM images

We review the manner in which lens aberrations, partial spatial coherence, and partial temporal coherence affect the formation of a sub-Å electron probe in an aberration-corrected transmission electron microscope. Simulations are used to examine the effect of each of these factors on a STEM image. It is found that the effects of partial spatial coherence (resulting from finite effective source size) are dominant, while the effects of residual lens aberrations and partial temporal coherence produce only subtle changes from an ideal image. We also review the way in which partial spatial and temporal coherence effects are manifest in a Ronchigram. Finally, we provide a demonstration of the Ronchigram method for measuring the effective source distribution in a probe aberration-corrected 300 kV field-emission gun transmission electron microscope.     

Multiplexing optical fiber low coherence and high coherence interferometric system with large range and high resolution for on-line measurement

A novel optical fiber sensing system multiplexing low coherence interferometry and high coherence interferometry that is endowed with large range and high resolution and is stabilized for on-line measurement is presented. An optical fiber Michelson interferometer performing measurement task in the system works in both modes of low coherence interferometry and high coherence interferometry simultaneously by employing a broadband light source and a fiber Bragg grating as an in-fiber reflective mirror. The amplitude of the measurand is determined by the low coherence interferometry while the value of the measurand is measured by the high coherence interferometry. Another optical fiber Michelson interferometer which is incorporated with the one performing measurement task stabilizes the sensing system for on-line measurement by exploiting an electronic feedback loop to reduce the influences that are resulted from environmental disturbances. The measurement range is 6 mm and the measurement uncertainty is less than 2 nm.     

Application of image processing to STEM tomography of low-contrast materials

In this study, the effect of various image-processing techniques on the visibility of tomographic reconstructions is investigated for a low-contrast material system of non-uniform thickness containing complex features such as grain boundaries and nanoparticles. Starting with a tilt series of high-angle annular dark-field (HAADF) images from an area of Dy-doped YBa₂Cu₃O_7−x-coated superconductor obtained using a scanning transmission electron microscope, various image-processing techniques were applied. These can be classified as edge detection, contrast-enhancing methods for non-uniform thickness and image sharpening. Although the processing methods violate the projection criterion for tomographic reconstruction, they were found, at least in this case, to enhance contrast and define the correct shape and size of structural features with minimal artifacts. Enhancing the visibility of structural features in this way allows the spatial distribution of the nanoparticles, their size, number density and location relative to each other and grain boundaries to be determined, which are essential to understand the flux-pinning characteristics of these materials.     

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).     

Practical methods for the measurement of spatial coherence--a comparative study

Two new methods for the measurement of transverse spatial coherence in a transmission electron microscope (TEM) are developed and applied to measure the spatial coherence in a field emission gun TEM. Measurements are made under different illumination and operating conditions, illustrating the effect of these conditions on the spatial coherence. The relative merits and limitations of these methods are discussed and compared, together with the previously described “Ronchigram” method.     

High-resolution wave field reconstruction using a through-focus series of images taken under limited electron dose

The three-dimensional Fourier filtering method and Schiske's Wiener filtering method are compared with the aim of high-resolution wave field reconstruction of an unstained deoxyribonucleic acid (DNA) molecular fiber using a through-focus series of images taken under a limited electron dose. There were some definite differences between the two reconstructed images, although the two kinds of processing are essentially equivalent except for the dimension and the filter used for processing. Through theoretical analyses together with computer simulations, the differences were proved to be primarily due to specimen drift during the experiment. Although the observed structure of the DNA molecular fiber was heavily damaged by electron beam irradiation, reconstructed images by the three-dimensional Fourier filtering method provided higher resolution information on the molecular structure even when relatively large specimen drift was included in the through-focus series. In contrast, in Schiske's Wiener filtering method, the detailed information of the structure was lost because of the drift, although the reconstructed image showed a higher signal-to-noise ratio. The three dimensional Fourier filtering method seems to be more applicable for observing radiation-sensitive materials under an extremely low electron dose, because specimen drift cannot be completely avoided.     

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.     

Rapid in‐focus corrections on quantitative amplitude and phase imaging using transport of intensity equation method

Transport of intensity equation (TIE) method can acquire sample phase distributions with high speed and accuracy, offering another perspective for cellular observations and measurements. However, caused by incorrect focal plane determination, blurs and halos are induced, decreasing resolution and accuracy in both retrieved amplitude and phase information. In order to obtain high‐accurate sample details, we propose TIE based in‐focus correction technique for quantitative amplitude and phase imaging, which can locate focal plane and then retrieve both in‐focus intensity and phase distributions combining with numerical wavefront extraction and propagation as well as physical image recorder translation. Certified by both numerical simulations and practical measurements, it is believed the proposed method not only captures high‐accurate in‐focus sample information, but also provides a potential way for fast autofocusing in microscopic system.     

Spatial incoherence in phase retrieval based on focus variation

We investigate the effect of spatial incoherence on two methods of phase retrieval based on focus variation: the transport of intensity equation and iterative wave function reconstruction. Spatial incoherence provides an upper bound on the defocus step size which should be used in each case. The requirement that phase information manifests itself in sufficient variation in the defocused images provides a lower bound on the defocus step size which should be used in each case. The scaling of these upper and lower bounds with object size and imaging resolution differs in such a way that, given the spatial incoherence properties of the source, for sufficiently low resolutions neither technique can retrieve phase information. The regions of applicability of the two techniques are discussed.     

Comparing electron tomography and HRTEM slicing methods as tools to measure the thickness of nanoparticles

Nanoparticles’ morphology is a key parameter in the understanding of their thermodynamical, optical, magnetic and catalytic properties. In general, nanoparticles, observed in transmission electron microscopy (TEM), are viewed in projection so that the determination of their thickness (along the projection direction) with respect to their projected lateral size is highly questionable. To date, the widely used methods to measure nanoparticles thickness in a transmission electron microscope are to use cross-section images or focal series in high-resolution transmission electron microscopy imaging (HRTEM “slicing”). In this paper, we compare the focal series method with the electron tomography method to show that both techniques yield similar particle thickness in a range of size from 1 to 5 nm, but the electron tomography method provides better statistics since more particles can be analyzed at one time. For this purpose, we have compared, on the same samples, the nanoparticles thickness measurements obtained from focal series with the ones determined from cross-section profiles of tomograms (tomogram slicing) perpendicular to the plane of the substrate supporting the nanoparticles. The methodology is finally applied to the comparison of CoPt nanoparticles annealed ex situ at two different temperatures to illustrate the accuracy of the techniques in detecting small particle thickness changes.