Spatial Frequency in EM Imaging
- Practical Electron Microscopy and Database -
- An Online Book -

http://www.globalsino.com/EM/  



 

This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.

 

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The image of every electron lens in EMs (electron microscopes) is mathematically formed by a Fourier transformation from the lens front to the back focal plane. That means the spatial coordinates are transformed into their reciprocal values, being spatial frequency spectra theoretically. Those lenses are also called Fourier lens, Fourier transform lens, or Fourier transformation lens, reflecting the two dimensional Fourier transformation takes place. Note that small distances in the object (e.g. between atomic planes) correspond to large spatial frequencies and vice versa.

The third- and fifth-order spherical aberration coefficients and defocus are present in a perfect, round electromagnetic lens, while all other higher-order coherent aberration coefficients in Table 3740 are caused by lens imperfections. Those higher-order aberration coefficients can become important at high spatial resolutions because of the increasing order of their spatial frequency dependence.

FEG-TEMs with field-emission electron sources [1 - 2] offering much better spatial and temporal coherence than those offered by CTEMs with LaB6 and W (tungsten) electron guns. Therefore, FEG-TEMs have allowed images to contain much higher spatial frequencies [3 - 4]. Note that for a perfectly coherent illumination, the resulting phase contrast transfer function continues to oscillate between +1 and -1 even at very large spatial frequencies.

For real TEMs, the phase contrast transfer function is limited by a number of factors: i) Variation in electrical currents in the lenses; ii) Variation in accelerating voltage and gun emission; iii) Attenuation in the CCD camera; iv) Slightly nonparallel illumination; and v) Specimen drift and vibration. All these limits can be described by envelope functions, E, which will gradually dampen the phase contrast transfer function and eventually reduce it below the noise at some spatial frequency. The first three factors depend on the inherent properties of the microscope. For all modern microscopes this significant decrease happens at higher spatial frequencies than the point to point resolution and defines the information limit of the microscope beyond which no information is transferred at all. In other words, the information limit is defined as the highest spatial frequency transferred to the image with statistical significance.

 

 

[1] M.T. Otten, W.M.J. Coene, Ultramicroscopy 48 (1–2) (1993) 77–91.
[2] T. Honda, T. Tomita, T. Kaneyama, Y. Ishida, Ultramicroscopy 54 (2–4) (1994) 132–144.
[3] W. Coene, G. Janssen, M. Op de Beeck, D. van Dyck, Phys. Rev. Lett. 69 (1992) 3743–3747.
[4] M. Op de Beeck, D. van Dyck, W. Coene, Ultramicroscopy 64 (1–4) (1996) 167–183.

 

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