Analytic integration of contrast transfer functions

1988 ◽  
Vol 46 ◽  
pp. 822-823
Author(s):  
Peter Rez
Keyword(s):  
Wave Function ◽  
Transfer Function ◽  
Transfer Functions ◽  
Spherical Aberration ◽  
Continuous Distribution ◽  
Objective Lens ◽  
Direction Parallel ◽  
Scattering Angle ◽  
Analytic Integration ◽  
Image Position

In high resolution microscopy the image amplitude is given by the convolution of the specimen exit surface wave function and the microscope objective lens transfer function. This is usually done by multiplying the wave function and the transfer function in reciprocal space and integrating over the effective aperture. For very thin specimens the scattering can be represented by a weak phase object and the amplitude observed in the image plane is1where fe (Θ) is the electron scattering factor, r is a postition variable, Θ a scattering angle and x(Θ) the lens transfer function. x(Θ) is given by2where Cs is the objective lens spherical aberration coefficient, the wavelength, and f the defocus.We shall consider one dimensional scattering that might arise from a cross sectional specimen containing disordered planes of a heavy element stacked in a regular sequence among planes of lighter elements. In a direction parallel to the disordered planes there will be a continuous distribution of scattering angle.

Electron Holography Improving Transmission Electron Microscopy

1990 ◽  
Vol 48 (1) ◽  
pp. 208-209
Author(s):  
Hannes Lichte
Keyword(s):  
Electron Microscopy ◽  
Transfer Functions ◽  
Imaging System ◽  
Spherical Aberration ◽  
Image Plane ◽  
Chromatic Aberration ◽  
Object Wave ◽  
Objective Lens ◽  
Electron Holography ◽  
Wave Aberration

Generally, the electron object wave o(r) is modulated both in amplitude and phase. In the image plane of an ideal imaging system we would expect to find an image wave b(r) that is modulated in exactly the same way, i. e. b(r) =o(r). If, however, there are aberrations, the image wave instead reads as b(r) =o(r) * FT(WTF) i. e. the convolution of the object wave with the Fourier transform of the wave transfer function WTF . Taking into account chromatic aberration, illumination divergence and the wave aberration of the objective lens, one finds WTF(R) = Echrom(R)Ediv(R).exp(iX(R)) . The envelope functions Echrom(R) and Ediv(R) damp the image wave, whereas the effect of the wave aberration X(R) is to disorder amplitude and phase according to real and imaginary part of exp(iX(R)) , as is schematically sketched in fig. 1.Since in ordinary electron microscopy only the amplitude of the image wave can be recorded by the intensity of the image, the wave aberration has to be chosen such that the object component of interest (phase or amplitude) is directed into the image amplitude. Using an aberration free objective lens, for X=0 one sees the object amplitude, for X= π/2 (“Zernike phase contrast”) the object phase. For a real objective lens, however, the wave aberration is given by X(R) = 2π (.25 Csλ3R4 + 0.5ΔzλR2), Cs meaning the coefficient of spherical aberration and Δz defocusing. Consequently, the transfer functions sin X(R) and cos(X(R)) strongly depend on R such that amplitude and phase of the image wave represent only fragments of the object which, fortunately, supplement each other. However, recording only the amplitude gives rise to the fundamental problems, restricting resolution and interpretability of ordinary electron images:

High-resolution electron holography of MgO

1991 ◽  
Vol 49 ◽  
pp. 672-673
Author(s):  
T. Tanji ◽  
K. Urata ◽  
K. Ishizuka
Keyword(s):  
Image Processing ◽  
Wave Function ◽  
Spherical Aberration ◽  
Objective Lens ◽  
Electron Holography ◽  
Electron Wave Function ◽  
Resolution Limit ◽  
Electron Wave ◽  
Transmission Electron ◽  
Resolution Electron

Electron holography is a useful application of a transmission electron microscope instrument equipped with a field emission gun (FE-TEM). The peculiarity of holography is ability to record and reconstruct the complex amplitude of an electron wave function. This characteristic makes many kinds of image processing applicable, for instance, image restoration and interferometry. Especially the correction of aberrations is expected to overcome the resolution limit owing to the spherical aberration of an electron objective lens. A few preliminary works have been reported, where a laser optical system or a digital computer system was used to reconstruct image waves and to correct the aberrations. The image qualities, however, were not enough to improve the point resolution.

Determination of Energy Spread and Source Size by Optical Diffraction

1975 ◽  
Vol 33 ◽  
pp. 182-183
Author(s):  
J. Frank
Keyword(s):  
Transfer Function ◽  
Spherical Aberration ◽  
Careful Analysis ◽  
Energy Spread ◽  
Optical Diffraction ◽  
Source Distribution ◽  
Contrast Transfer Function ◽  
Spatial Frequencies ◽  
Gaussian Source ◽  
Image Position

Since the effect of energy spread and angular spread of illumination on the contrast transfer function is known from theoretical investigations, these experimental parameters can in turn be obtained by careful analysis of electron micrographs. For Gaussian energy spread (halfwidth △E), negligible voltage and current fluctuations and a Gaussian source distribution (half-width α), the coherent transfer function for monochromatic electrons appears multiplied with the envelope function(θ scattering angle, λ electron wavelength, E0 energy, △z defocus; Cc, Cs chromatic and spherical aberration constants). This causes an attenuation of the image Fourier transform at high spatial frequencies and is responsible for the fact that the visible bandlimit in optical diffraction patterns is well below the aperture limit.

Atomic resolution from 500kv electron micrographs by computer image processing

1978 ◽  
Vol 36 (1) ◽  
pp. 220-221
Author(s):  
N. Uyeda ◽  
E. Kirkland ◽  
Y. Fujiyoshi ◽  
B. Siegel
Keyword(s):  
Additive Noise ◽  
Spherical Aberration ◽  
Objective Lens ◽  
Bright Field Image ◽  
Bright Field ◽  
Contrast Transfer Function ◽  
Signal Ratio ◽  
Computer Image Processing ◽  
The Fourier Transform ◽  
Image Position

We have successfully applied the method suggested by P. Schiske (1) for optimally combining several micrographs of different defocus values (Δfβ) in the presence of additive noise. If c(k) represents the Fourier transform of tne ideal, unaberrated bright field image, and jβ(k) the transform of the real micrographs obtained at different defocus values, then the filter functions Qβ(k) that reducesto a minimum, was shown (1) to be, within a common multiplicative factor,The contrast transfer function (CTF) is where λ is the electron wavelength and Cs is the spherical aberration coefficient of the objective lens. η(k) is the noise to signal ratio in reciprocal space for unit CTF.This procedure was applied to images of a tilted crystaline specimen of hexadecachloro- phtalocyanatocopper (II) taken on the Kyoto 500KV electron microscope (2).

On the magnetic electron lens of minimum sphrical aberration

1978 ◽  
Vol 36 (1) ◽  
pp. 24-25
Author(s):  
S. Suzuki ◽  
A. Ishikawa
Keyword(s):  
Spherical Aberration ◽  
Resolving Power ◽  
Objective Lens ◽  
Lens Design ◽  
Magnetic Lens ◽  
Initial Voltage ◽  
Path Distance ◽  
Electron Lens ◽  
Image Position ◽  
Magnetic Electron

For the development of the electron microscope, in which high resolving power is demanded, it is important to construct an electron objective lens with minimum spherical aberration.In 1943, one of the authors published the paper on the approximate calculation of the electromagnetic field to give a minimum spherical aberration and also published the papers on small spherical aberration lens design based on this calculation.We will speak a comparison between the experimental results and the numerical calculations in practical cases.The following line shows the method to get more strictly minimum spherical aberration of magnetic lens.In a space charge free electron optical system, where a pure magnetic lens is concerned, differential equation for paraxial electron path is given byU being the initial voltage applied to the electron beam and γ the path distance from the optical axis Z.

Digital Image Processing of High Resolution 500KV Micrographs

1981 ◽  
Vol 39 ◽  
pp. 234-235
Author(s):  
E.J. Kirkland ◽  
B.M. Siegel ◽  
N. Uyeda ◽  
Y. Fujiyoshi
Keyword(s):  
Phase Contrast ◽  
Transfer Functions ◽  
Spherical Aberration ◽  
Random Noise ◽  
Image Plane ◽  
Partial Coherence ◽  
Thin Specimen ◽  
The Fourier Transform ◽  
Image Position ◽  
Contrast Image

The predominate linear components of a defocus series of bright field phase contrast electron micrographs of a thin specimen may be represented in Fourier space as;(1)If there are m micrographs in the series then are m dimensional vectors, each component of which represents one micrograph. is the Fourier transform of the recorded defocus series and is the random noise content of is an mx2 matrix representing the transfer functions of the microscope (including spherical aberration, defocus, and partial coherence). is a two component vector representing the two components (real and imaginary) of the ideal unaberrated phase contrast image, is a two dimensional spatial frequency vector in the image plane.

Atomic Structure Determination Using the Focal-Series Reconstruction Technique

2001 ◽  
Vol 7 (S2) ◽  
pp. 286-287
Author(s):  
A. Thust ◽  
C.L. Jia
Keyword(s):  
Wave Function ◽  
High Resolution ◽  
Atomic Structure ◽  
Materials Science ◽  
Spherical Aberration ◽  
Objective Lens ◽  
Reconstruction Technique ◽  
Exit Plane ◽  
Reconstruction Procedure ◽  
Starting Point

During the last five years the technique focal-series reconstruction has evolved to a powerful tool for investigating materials science problems in high-resolution transmission electron microscopy. Compared to the conventional interpretation of one single high-resolution image, the quantum mechanical electron wave function at the exit plane of the object (exit-plane wave function, EPW) is in many cases a better starting point for the materials analysis. The retrieval of the EPW is achieved on a routine basis by applying automated numerical procedures to a series of images taken from the same specimen area at different objective lens defocus values. The application of the reconstruction procedure allows one to remove numerically all the well known instrumental artifacts, such as nonlinear contrast formation or delocalisation effects due to spherical aberration and other parasitic aberrations. The reconstructed EPW gives thus direct insight into the atomic structure in the case of sufficiently thin objects and renders tedious image simulations of complicated defects unnecessary in many cases.

A Calibration Method Usable in the Production of Phase Plates for High Resolution Electron Microscopy

1971 ◽  
Vol 29 ◽  
pp. 46-47
Author(s):  
F. Thon ◽  
D. Willasch
Keyword(s):  
Electron Microscopy ◽  
Frequency Spectrum ◽  
Spherical Aberration ◽  
High Resolution Electron Microscopy ◽  
Calibration Method ◽  
Objective Lens ◽  
Resolution Electron ◽  
Diffraction Angle ◽  
Phase Plates ◽  
Image Position

It is well known that phase contrast in electron microscopy is normally due to the phase shift γ introduced by defocusing and spherical aberration of the objective lens. It is given by Scherzer's formula (2) :(1)where γ is the electron wavelength, Cs the spherical aberration coefficient, θ the diffraction angle and Δz the defocus value.However, in this way, only sections out of the spatial frequency spectrum contained in the abject are transferred to the image. In order to transmit the whole frequency spectrum, it has been proposed that a phase shifting plate is inserted into the back focal plane of the objective lens.

EM contrast transfer functions for tilted illumination imaging

1978 ◽  
Vol 36 (1) ◽  
pp. 168-169
Author(s):  
Ondrej L. Krivanek
Keyword(s):  
Electron Microscopy ◽  
Transfer Function ◽  
Transfer Functions ◽  
Theoretical Description ◽  
Phase Object ◽  
Contrast Transfer Function ◽  
Transfer Theory ◽  
Weak Phase ◽  
Theoretical Predictions ◽  
Image Position

The transfer theory of electron microscopy has now been firmly established, at least for the case of axial illumination. Tilted illumination introduces slight complications, but even so the theoretical description of the contrast transfer remains essentially quite simple. Experimentally, the existence of the main "achromatic circle" and the subsidiary circles due to plasmon scattering has been well demonstrated, but the theoretical predictions for the change in the apparent defocus and astigmatism,as determined from the image modified by tilting, has not been confirmed. This paper aims to close the gap.With coherent illumination, the contrast transfer from a weak phase object and for tilted illumination is described by including the beam tilt in the familiar axial expressions:where h(q) is the contrast transfer function (CTF), X(q) the aberration function, and the remaining symbols have their usual meaning. The derivation is algebraically lengthy, but the results can be summarised as follows.

Computer simulation of phase of contrast of cluster atoms on support film made by sputter deposition

1978 ◽  
Vol 36 (1) ◽  
pp. 234-235
Author(s):  
Tetsuo Oikawa ◽  
Fumiko Ishigaki ◽  
Shuhei Fukuyama ◽  
Haruo Wakizaka ◽  
Kiichi Hojou ◽  
...  
Keyword(s):  
Wave Function ◽  
Scattering Amplitude ◽  
Spherical Aberration ◽  
Gold Cluster ◽  
Sputter Deposition ◽  
Single Atom ◽  
Diffraction Integral ◽  
The Monte Carlo Method ◽  
Kirchhoff Diffraction Integral ◽  
Image Position

Based on the image formation theory, the wave function of a single atom consisting of the complex scattering amplitude |f(0)|exp{-inj(0)} and phase shift γj of spherical aberration is rigorously obtained by the Kirchhoff diffraction integral in the Fraunhofer approximation as a function of defocus. Using of the wave function w(υj,γj) of a group of atoms, which is extended the wave function of a single atom, variations of imaging phase contrast of tungsten support film in through focal series made by sputter deposition are simulated. The contrast of tungsten support film and a gold cluster atoms on the substrate made by sputter depposition, which are very close with the experiment, are calculated by the Monte-Carlo method with the aid of the inter-atomic size distribution measured by electron diffraction pattern.The contrast g for cluster atoms has been calculated by integrating the scattering wave for coherent waves, and for incoherent waves the amplitude scattered outside the limiting angle α set by the aperture respectively,

Sign in / Sign up

Export Citation Format

Share Document