Robert A. McLeod

Phase measurement error in summation of electron holography series.

Off-axis electron holography is a method for the transmission electron microscope (TEM) that measures the electric and magnetic properties of a specimen. The electrostatic and magnetic potentials modulate the electron wavefront phase. The error in measurement of the phase therefore determines the smallest observable changes in electric and magnetic properties. Here we explore the summation of a hologram series to reduce the phase error and thereby improve the sensitivity of electron holography. Summation of hologram series requires independent registration and correction of image drift and phase wavefront drift, the consequences of which are discussed. Optimization of the electro-optical configuration of the TEM for the double biprism configuration is examined. An analytical model of image and phase drift, composed of a combination of linear drift and Brownian random-walk, is derived and experimentally verified. The accuracy of image registration via cross-correlation and phase registration is characterized by simulated hologram series. The model of series summation errors allows the optimization of phase error as a function of exposure time and fringe carrier frequency for a target spatial resolution. An experimental example of hologram series summation is provided on WS2 fullerenes. A metric is provided to measure the object phase error from experimental results and compared to analytical predictions. The ultimate experimental object root-mean-square phase error is 0.006 rad (2π/1050) at a spatial resolution less than 0.615 nm and a total exposure time of 900 s. The ultimate phase error in vacuum adjacent to the specimen is 0.0037 rad (2π/1700). The analytical prediction of phase error differs with the experimental metrics by +7% inside the object and -5% in the vacuum, indicating that the model can provide reliable quantitative predictions.

Characterization of detector modulation-transfer function with noise, edge, and holographic methods

We developed a new method for characterization of detector performance used in the transmission electron microscope (TEM) based on the measured contrast of holographic fringes. The new method changes spatial frequency of the measured holographic fringes, generated by an electrostatic biprism and Schottky or cold field-emission gun, to sample the modulation-transfer function (MTF) of the detector. The MTF of a Gatan Ultrascan™ 1000 charged-coupled detector (CCD) is evaluated using the new method and the results are compared to the established noise and slanted-edge method results. Requirements for accuracy of the edge and noise MTF methods are discussed. We consider issues surrounding incomplete read-out and how it affects the gain reference normalization of the detector. We evaluate how the MTF affects optimization of experimental parameters in the TEM.

Validity of the dipole approximation in TEM-EELS studies.

Nondipole effects in electron energy‐loss spectroscopy are evaluated in terms of deviation of the inelastic scattering from a Lorentzian angular distribution, which is assumed in established procedures for plural‐scattering deconvolution, thickness measurement, and Kramers‐Kronig analysis. The deviation appears to be small and may be outweighed by the effect of plural (elastic + inelastic) scattering, which is not removed by conventional deconvolution methods. In the core‐loss region of the spectrum, non‐Lorentzian behaviour stems from a reduction of the generalized oscillator strength from its optical value and (for energies far above an ionization threshold) formation of a Bethe‐ridge angular distribution. At incident energies above 200 keV, retardation effects further distort the angular dependence, even for core losses just above threshold. With an on‐axis collection aperture, non‐dipole effects are masked by the rapid falloff of intensity with scattering angle, but they may become important for off‐axis measurements. Near‐edge fine structure is sensitive to nondipole effects but these can be minimized by use of an angle‐limiting collection aperture. Microsc. Res. Tech. 77:773–778, 2014.

Determination of localized visibility in off-axis electron holography.

Off-axis electron holography is a wavefront-split interference method for the transmission electron microscope that allows the phase shift and amplitude of the electron wavefront to be separated and quantitatively measured. An additional, third component of the holographic signal is the coherence of the electron wavefront. Historically, wavefront coherence has been evaluated by measurement of the holographic fringe visibility (or contrast) based on the minimum and maximum intensity values. We present a method based on statistical moments is presented that allows allow the visibility to be measured in a deterministic and reproducible fashion suitable for quantitative analysis. We also present an algorithm, based on the Fourier-ratio method, which allows the visibility to be resolved in two-dimensions, which we term the local visibility. The local visibility may be used to evaluate the loss of coherence due to electron scattering within a specimen, or as an aid in image analysis and segmentation. The relationship between amplitude and visibility may be used to evaluate the composition and mass thickness of a specimen by means of a 2-D histogram. Results for a selection of elements (C, Al, Si, Ti, Cr, Cu, Ge, and Au) are provided. All presented visibility metrics are biased at low-dose conditions by the presence of shot-noise, for which we provide methods for empirical normalization to achieve linear response.

Technique for Complex Averaging of Electron Holograms

Electron holography has been used to characterize electrostatic and magnetic behaviour of specimens in transmission electron microscopy [1]. Although widely applied, it remains a challenge to obtain the signal-to-noise ratio required to achieve the high phase resolution needed for observations of small variations in local charge on a specimen. Holography often requires long acquisition times implying the need for stable laboratory conditions. One approach to improve phase resolution is to combine many individual holograms acquired with short individual acquisition times and average them [2]. Here we report on the practical aspects of registration, alignment, and averaging of hologram series.

Determination of Camera Modulation-Transfer Function by Electron Holography

For sensor systems composed of arrays of pixels, it is often desirable to measure the degree of signal spread from a point source to nearby pixels as this results in a loss of contrast. The point-spread function (PSF) is a measure of the information spread from a single electron. The reciprocal space equivalent of the PSF is the modulation-transfer function (MTF), which represents the observed contrast at a given spatial (pixel) frequency. For transmission electron microscopy (TEM), electrons are typically diffracted from periodic structures so MTF is the preferred presentation.