Electron holography of long-range electric and magnetic fields

Abstract

The influence of the perturbed reference wave in electron holography is considered for the case of static electromagnetic microfields, whose extension around the observed specimen cannot be neglected. These microfields are called ‘‘long-range’’ to distinguish them from the ‘‘short-range’’ ones, whose extension is strictly limited within the object wave and hence do not perturb the reference wave. Optical reconstructions of experimental holograms of simple electrostatic or magnetic long-range fields have been modeled and simulated. The results indicate that perturbation effects must be taken into account when long-range microfields are investigated by electron holography; it is also shown that their influence can be minimized by increasing the interference distance between the object and reference wave.

Model-independent measurement of the charge density distribution along an Fe atom probe needle using off-axis electron holography without mean inner potential effects

“…It is also affected by perturbation of the vacuum reference wave, 20,21 as a result of the fact that an off-axis electron hologram is formed by overlapping a specimen electron wave with a reference wave that is affected by phase modulations associated with the electromagnetic field of the specimen itself. Here, the perturbed reference wave was taken into account in the model by rewriting the total phase shift / T in the form…”

“…A key advantage of the present approach, which is based on integration of the Laplacian of the phase or, equivalently, contour integration of the gradient of the phase, is that it is insensitive to perturbation of the vacuum reference wave used to generate the hologram by the electromagnetic field of the specimen itself, 19 so long as the region from which the reference wave is obtained is itself charge-free. [20][21][22] However, in the past, it has suffered from an important limitation resulting from the effect of the mean inner potential contribution to the phase shift on the measured charge density distribution. 23 We show that this limitation can be overcome by analysing the difference between electron holographic phase images acquired with different voltages applied to the specimen.…”

Towards quantitative off-axis electron holographic mapping of the electric field around the tip of a sharp biased metallic needle

“…However, a projection does not coincide with a cross-section of the potential being projected, unless the potential does not depend on the direction along which it is projected, i.e., V ¼ V(x,y); (ii) Whenever the potential is not strictly bounded within a finite domain, its tails will perturb the region where the vacuum reference wave travels, making it non-planar. [17][18][19] Here, as a result of the fact that the vacuum reference wave is perturbed by the stray field from the tip, rather than retrieving the ideal object wavefunction wðx; yÞ ¼ Aðx; yÞe i½uðx;yÞ ;…”

Interpretation of Holographic Contour Maps of Reverse Biased p-n Junctions

“…satisfies the boundary condition, (4), which in parabolic coordinates becomes Putting S = 0 for simplicity, this leaves us to determine only the function R, and this can be done by comparing our expression with the one which can be deduced from Erdelyi's formula (11) in Section 8.5.2 of the reference [8], specifying there c = -1/2, t = 1 and y = 0…”

“…Nonetheless, it has been the analysis of the p-n junction holographic interference maps and of the troubles associated in their processing and interpretation which prompted the investigations on long range electromagnetic fields [4], whose peculiar feature is represented by the fact that the so-called reference wave is in reality perturbed by the tail of the fringing field.…”