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).     

Restoration of weak phase-contrast images recorded with a high degree of defocus: the "twin image" problem associated with CTF correction

Relatively large values of objective-lens defocus must normally be used to produce detectable levels of image contrast for unstained biological specimens, which are generally weak phase objects. As a result, a subsequent restoration operation must be used to correct for oscillations in the contrast transfer function (CTF) at higher resolution. Currently used methods of CTF correction assume the ideal case in which Friedel mates in the scattered wave have contributed pairs of Fourier components that overlap with one another in the image plane. This "ideal" situation may be only poorly satisfied, or not satisfied at all, as the particle size gets smaller, the defocus value gets larger, and the resolution gets higher. We have therefore investigated whether currently used methods of CTF correction are also effective in restoring the single-sideband image information that becomes displaced (delocalized) by half (or more) the diameter of a particle of finite size. Computer simulations are used to show that restoration either by "phase flipping" or by multiplying by the CTF recovers only about half of the delocalized information. The other half of the delocalized information goes into a doubly defocused "twin" image of the type produced during optical reconstruction of an in-line hologram. Restoration with a Wiener filter is effective in recovering the delocalized information only when the signal-to-noise ratio (S/N) is orders of magnitude higher than that which exists in low-dose images of biological specimens, in which case the Wiener filter approaches division by the CTF (i.e. the formal inverse). For realistic values of the S/N, however, the "twin image" problem seen with a Wiener filter is very similar to that seen when either phase flipping or multiplying by the CTF is used for restoration. The results of these simulations suggest that CTF correction is a poor alternative to using a Zernike-type phase plate when imaging biological specimens, in which case the images can be recorded in a close-to-focus condition, and delocalization of high-resolution information is thus minimized.     

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.     

Direct structure inversion from exit waves: Part I: Theory and simulations

In order to interpret the amplitude and phase of the exit wave in terms of mass and position of the atoms, one has to “invert” the dynamic scattering of the electrons in the object so as to obtain a starting structure which can then be used as a “seed” for further quantitative structure refinement. This is especially challenging in case of a zone axis condition when the interaction of the electrons with the atom column is very strong. Based on the channelling theory we will show that the channelling map not only yields a circle on the Argand plot but also a circular “defocus curve” for every column. The former gives the number of atoms in each column, while the latter provides the defocus value for each column, which reveals the surface roughness at the exit plane with single atom sensitivity.     

Effect of a physical phase plate on contrast transfer in an aberration-corrected transmission electron microscope

In this theoretical study we analyze contrast transfer of weak-phase objects in a transmission electron microscope, which is equipped with an aberration corrector (C(s)-corrector) in the imaging lens system and a physical phase plate in the back focal plane of the objective lens. For a phase shift of pi/2 between scattered and unscattered electrons induced by a physical phase plate, the sine-type phase contrast transfer function is converted into a cosine-type function. Optimal imaging conditions could theoretically be achieved if the phase shifts caused by the objective lens defocus and lens aberrations would be equal to zero. In reality this situation is difficult to realize because of residual aberrations and varying, non-zero local defocus values, which in general result from an uneven sample surface topography. We explore the conditions--i.e. range of C(s)-values and defocus--for most favourable contrast transfer as a function of the information limit, which is only limited by the effect of partial coherence of the electron wave in C(s)-corrected transmission electron microscopes. Under high-resolution operation conditions we find that a physical phase plate improves strongly low- and medium-resolution object contrast, while improving tolerance to defocus and C(s)-variations, compared to a microscope without a phase plate.     

Automatic focus correction for spot-scan imaging of tilted specimens.

The variation in defocus within an image of a highly tilted specimen can be a serious source of artifact. Spot-scan imaging can be combined with dynamic focusing to greatly reduce this range of defocus. A protocol is described for determining the parameters required for the automatic focus compensation during the recording of a spot-scan image. Images of a gold test specimen demonstrate the efficacy of this procedure in extending the area of the image that contains high-quality data. In case the tilt angle or resolution is high enough that the height difference of the specimen within each small illuminated area is larger than the depth of field, the image must be treated to compensate for the focus variation. The same principle is used as was developed for compensation of conventional images of tilted specimens.     

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.     

Restraint of tool path ripple based on the optimization of tool step size for sub-aperture deterministic polishing

Tool path ripple error (TPR_error) is one of the main reasons due to the medium-high spatial frequency error on the surface of aspheric optics. The purpose of this paper is to analyze the effect of the tool step size to the TPR_error in sub-aperture deterministic polishing (SDP) and study a method which can optimize the tool step size to restrain this error. Three groups of simulation experiments were conducted using three different tool influence functions to simulate the uniform removal of the material. As the TPR_error is influenced by three factors, which are full width at half maximum (FWHM) of tool influence function (TIF), tool step size, and depth of material removed, each group of the experiments was conducted under the fixed TIF and depth of material removed. It turns out that both peak-to-valley (PV) and root-mean-square (RMS) values of the TPR_error become larger with the increase of the tool step size, and the variation tendency likes a reversed “L” shape curve. And, the method adopted in the simulation was further validated by the experiment. Therefore, the tool step size at the inflection point would be optimal to restrain the TPR_error together with saving the polishing time to a certain extent. This method could be used to determine the best-suited tool step size in SDP whose typical TIF is a Gaussian or Gaussian-like shape.     

基于时间散焦分析法的三维信息采集

为了测量被测物的距离信息,结合投影系统的三维信息采集方法,研究了一种基于时间散焦分析的三维信息采集方法.通过向被测物投影移动的条纹,分析照明模式在各点的散焦度,利用傅里叶变换解析被测物各点的距离信息.实验结果表明,该方法不需要考虑点匹配问题,能单独计算各点的位置信息,并且能测量较大的距离.     

HREM image simulations for small particle catalysts on crystalline supports

The imaging conditions for electron microscope studies of supported ultrafine particle catalysts have been investigated by multislice simulations. Images of Pt and ReO₄ particles ranging from 0·4 to 2·3 nm in size were simulated in both plan view and profile view with a rutile (TiO₂) support. It was shown that particle visibility varied greatly with the objective lens defocus. Optimum defocus was not favourable for supported particles in plan view since the ultrafine supported particles were least visible at this defocus. Underfocusing, especially at defoci corresponding to half-spacing fringes in the TiO₂ support, led to improved visibility and resolution of the supported particles. Although the structure and shape of supported ultrafine particles should be resolved better with a 400-kV high-resolution electron microscope, their detectability is poorer than with a 200-kV instrument. An ReO₄ cluster should be detectable at 200 kV on TiO₂ supports up to 5 nm in thickness, whereas it is only likely to be detectable at 400 kV on supports up to 3 nm in thickness. The simulations confirmed that optimum defocus is most favourable for imaging supported particles in profile view. Atomic information for particles as small as a 13-atom Pt cuboctahedral cluster should be resolvable with a 400-kV instrument. The crystalline Ti monolayer observed on surfaces of Pt particles, which could explain the mechanism known as SMSI, was simulated as an example of profile imaging.     

Quantitative Fresnel Lorentz microscopy and the transport of intensity equation

Imaging of the magnetic structure of thin films by the Fresnel mode of Lorentz microscopy has been re-evaluated in terms of the Ampérian current density within a sample. The conditions for which this imaging can be treated as linear are discussed for quantitative application. Additionally, the consequences for magnetic phase reconstruction using the transport of intensity equation for defocused images are considered. While the range of applicability may initially appear rather limited examples of objects containing different spatial frequency components are used to illustrate the possibilities where relatively large defocus values may be used.     

Defocus estimation from stroboscopic cryo-electron microscopy data

Defocus estimation is an important step for improving the resolution of single particle reconstructions. It can be troublesome to estimate the defocus from low-dose cryo-electron microscopy (cryo-EM) data, particularly if there is not sufficient contrast present in the Fourier transform of the micrograph. Most existing approaches estimate the defocus from the presence of Thon rings within the power spectrum, employing image enhancement techniques to highlight these rings. In this paper, an approach to estimating the defocus from a stroboscopic image series is described. The image series is used to obtain two statistical metrics: figure of merit (FOM) and Q-factor. These metrics have been used to estimate the defoci from low-dose stroboscopic cryo-EM data consisting of a variable number of images.     

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.     

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.     

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.     

Invited review article: measurement uncertainty of linear phase-stepping algorithms

Phase retrieval techniques are widely used in optics, imaging and electronics. Originating in signal theory, they were introduced to interferometry around 1970. Over the years, many robust phase-stepping techniques have been developed that minimize specific experimental influence quantities such as phase step errors or higher harmonic components of the signal. However, optimizing a technique for a specific influence quantity can compromise its performance with regard to others. We present a consistent quantitative analysis of phase measurement uncertainty for the generalized linear phase stepping algorithm with nominally equal phase stepping angles thereby reviewing and generalizing several results that have been reported in literature. All influence quantities are treated on equal footing, and correlations between them are described in a consistent way. For the special case of classical N-bucket algorithms, we present analytical formulae that describe the combined variance as a function of the phase angle values. For the general Arctan algorithms, we derive expressions for the measurement uncertainty averaged over the full 2π-range of phase angles. We also give an upper bound for the measurement uncertainty which can be expressed as being proportional to an algorithm specific factor. Tabular compilations help the reader to quickly assess the uncertainties that are involved with his or her technique.     

Influence of zero-loss filtering on electron optical phase contrast

Complex scattering amplitudes are used to calculate the phase contrast of colloidal gold particles. Comparison of measurements of the phase contrast intensity at the centre of the gold particle as a function of defocus for unfiltered and zero-loss filtered images demonstrates the increase in phase contrast achieved by zero-loss filtering even for a thick carbon substrate film. The granulation of amorphous germanium films is measured by the spatial rms (root mean square) values of image intensity in a defocus series.     

The theory of scanning microscopes with Gaussian pupil functions

The theory of imaging in scanning microscopes with lenses, source and detector all having Gaussian pupil function is developed. This assumption is useful as the expressions may be evaluated analytically. It is shown that Type 2 microscopes exhibit superior performance to those of Type 1. Effects of defocus are considered. It is found that defocus can be used in a Type 2 microscope to observe phase information without the limitation in resolution associated with stopping down the collector of a conventional microscope. It is also found that a Type 2 microscope discriminates against light scattered by parts of the object outside of the focal plane, allowing observation of detail within a thick object.