Phase interpolation Fourier transform beamforming for spatial division applications

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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FR-IR Molecular Microanalysis System

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100134958 ◽

1990 ◽

Vol 48 (2) ◽

pp. 270-271

Author(s):

John A. Reffner ◽

William T. Wihlborg

Keyword(s):

Fourier Transform ◽

Fourier Transform Infrared ◽

Integrated System ◽

Morphological Examination ◽

Fully Integrated ◽

Ir Microscopy ◽

Normal Functions ◽

Infrared Microspectroscopy ◽

Infrared Spectral ◽

Ft Ir

The IRμs™ is the first fully integrated system for Fourier transform infrared (FT-IR) microscopy. FT-IR microscopy combines light microscopy for morphological examination with infrared spectroscopy for chemical identification of microscopic samples or domains. Because the IRμs system is a new tool for molecular microanalysis, its optical, mechanical and system design are described to illustrate the state of development of molecular microanalysis. Applications of infrared microspectroscopy are reviewed by Messerschmidt and Harthcock.Infrared spectral analysis of microscopic samples is not a new idea, it dates back to 1949, with the first commercial instrument being offered by Perkin-Elmer Co. Inc. in 1953. These early efforts showed promise but failed the test of practically. It was not until the advances in computer science were applied did infrared microspectroscopy emerge as a useful technique. Microscopes designed as accessories for Fourier transform infrared spectrometers have been commercially available since 1983. These accessory microscopes provide the best means for analytical spectroscopists to analyze microscopic samples, while not interfering with the FT-IR spectrometer’s normal functions.

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The extended Fourier algorithm: Application in discrete optics and electron holography

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100172322 ◽

1994 ◽

Vol 52 ◽

pp. 918-919

Author(s):

E. Voelkl ◽

L. F. Allard

Keyword(s):

Fourier Transform ◽

Fourier Transforms ◽

Sampling Rate ◽

Electron Holography ◽

Optical Bench ◽

Fourier Space ◽

Ccd Cameras ◽

Simulated Image ◽

Initial Sampling ◽

Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

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