Spatial Coherence/Incoherence of Electron Sources
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The most promising coherent electron sources for electron microscopes (EMs) have been found in ultrasharp nanotips [1–3] in field emission electron guns due to the very high coherence and significant brightness. The important, common structural feature of their brightest field emitters is that they end with a single atom at the apex of a nanoprotrusion. The resulting characteristics are high coherence, strong focusing, sharp energy spectra, and high degeneracy [4] that significantly improve the spatial and temporal coherence properties. The use of ultrasharp nanotips in the EMs significantly improves the resolution and magnification, and allows for the simultaneous imaging of a larger area.

The coherence of an electron source reflects the phase differences in the emitted beam, and thus it determines the degree of interference that can occur between the direct and the diffracted waves. The phases of two electron beams can be correlated if they originate from a small electron source. The electron beams are coherent or partially coherent, respectively, depending on whether the correlation is complete or partial. The superposition of coherent or partially coherent beams leads to interference effects. Therefore, an interference pattern reflects the phase relation between the two waves from the two beams.

The coherence depends on both the origin of the electrons and their energy. The effect of the origin refers to spatial coherence or spatial incoherence, while the effect of the energy refers to temporal coherence or temporal incoherence. Therefore, electrons which originate from the same location but have different energies are temporally incoherent.

In ideal EMs (electron microscopies), e.g. in point-like monochromatic TEM (transmission electron microscopy), we can assume that the electron beam is generated from a point-like source and the energy variation ΔE is zero. This electron source is fully coherent. Even though the TEM specimen can be a crystal and thus, the incident electron beam k0 is split into different Bragg-diffracted components kg, resulting in a characteristic phase shift, the phase relation between the diffracted beams is maintained (the elastic electron scattering is coherent) [5]. This coherent relationship induces an interference pattern forming a HRTEM image. In real TEMs, the electron source has a finite size and produces electrons of slightly varying energies, so that the incident electron beam is not fully coherent but partially coherent. The finite size of the electron source results in partial spatial coherence and the small energy spread (ΔE ≠ 0) of the beam results in partial temporal coherence of the beam. Therefore, in reality, the electron source in EMs (electron microscopes) is neither spatially nor temporally perfectly coherent. However, cold field-emission (CFE) guns have high temporal coherence due to ΔE ≈ 0.3 eV and high spatial coherence because of the very small source size.

In EMs, the condenser aperture is used to exclude electrons emitted at high angles from the electron gun, which will decrease the brightness but improve the quality of the illumination because these peripheral electrons are less coherent, especially in LaB6 and W (tungsten) guns.

Information limit depends on the damping envelope incorporating partial temporal coherence due to chromatic aberration, but not partial spatial coherence due to beam convergence. Astigmatism in condenser lens is important because it reduces the coherence of the electron beam, while astigmatism in objective lens is important because it induces a serious degradation of spatial resolution.






[1] M. Rezeq, J. Pitters, R.A. Wolkow, J. Chem. Phys. 124 (2006) 204716.
[2] V.T. Binh, N. Garcia, Ultramicroscopy 42–44 (1992) 80.
[3] K. Nagaoka, H. Fujii, K. Matsuda, M. Komaki, Y. Murata, C. Oshima, T. Sakurai, Appl. Surf. Sci. 182 (2001) 12.
[4] V.T. Binh, S.T. Purcell, Appl. Surf. Sci. 111 (1997) 157.
[5] L. H. Schwartz and J. B. Cohen, (1987) Diffraction from Materials, 2nd ed., Berlin: Springer-Verlag.