Even though the original electron holography work described the reconstruction of an image by illuminating an ‘in-line’ electron hologram with a parallel beam, the reconstructed image is disturbed by a ‘ghost’ or ‘conjugate’ twin image. The method of electron holography that is most often used and is also most successfully performed for solving problems in materials science is the off-axis, or ‘sideband’, mode.
By far the most popular scheme of electron holography with high-energy electrons (80-200keV) in TEMs (transmission electron microscopes) is image-plane off-axis holography using a Möllenstedt–Düker biprism. In practice, a nearby hole in the TEM specimen allows free passage for a reference wave. And, both the object and reference waves are imaged
by the objective lens. The biprism deflects both waves towards each other at an overlapping angle (β') when a positive voltage is applied to the filament. Coherent superposition of the reference wave to the wave in the image plane forms off-axis image-plane holograms.
A small cable connects and applies a defined potential to the biprism filament. The biprism voltage is provided by highly stabilized DC power supplies. Practically, the charged wire in Figure 4308a splits the electron beam into two parts by very thin positively charged wire and acts to tilt a reference wave relative to an image-forming wave. Because of the non-uniform distribution of the electric fields around the wire the two parts of electrons interfere below the biprism such that the overlapping waves create an interference pattern of parallel fringes. These fringes are changed in position and contrast, depending upon how the specimen affects the electron beam. A digital hologram is recorded in a slightly defocused image plane of the object by a CCD camera system.
Based on numerical processing, reconstruction of off-axis holograms can be done by separating amplitude and phase images. For instance, the resulted interfere can be used to measure the phase shift of the electron wave which is sensitive to the electrostatic potential in the TEM specimen. The reconstruction process consists of two mathematical transformations:
i) Fourier transform of the hologram is performed. In this step, the resulting complex image is composed of the center band with autocorrelation and two mutually conjugated sidebands. Only one side band is chosen by applying a low-pass filter (normally round mask) centered on the selected side-band.
ii) The selected side-band is re-positioned to the center of the complex image and then the second (backward) Fourier-transform is applied. The resulting image presents complex values, reconstructing the amplitude and phase distributions of the object function.
Figure 4308a. Schematic diagram of off-axis electron holography in TEMs.
One partial electron wave becomes the reference wave by transmission through vacuum, and the other one generates the object exit wave by transmission through the specimen, given by,
Both parts of the wave are transferred by the objective lens to form the image wave,
Assuming the contrast of the interference fringes in the hologram is µ, the beam current of the hologram in the overlapping region can be given by ,
Ihol = - β*ln(µ)/k2 ----------------------- [4308c]
k -- The electron wave vector,
β -- The brightness of the electron source.
More accurately, Ihol can be given by, 
|qc| -- (=kβ'), the carrier frequency of the interference
β' -- The overlapping angle.
The fringe contrast is determined by the electron source coherence, the inelastic interaction, the instabilities, and the modulation transfer function (MTF) of the detector. The amplitude A(r) and the phase ϕ(r) are encoded as the contrast modulation and as the bending of the interference fringes, respectively. 
The charged wire can be made from a gold-coated quartz fiber at a diameter of 1 µm or less . This fiber is covered with a thin metal coating, e.g. gold (Au) and is mounted in the biprism assembly provided by manufacturers. The biprism can be either part of the selected area diffraction (SAD) aperture assembly (i.e. in place of one of the SAD apertures)  or a completely separate, retractable unit (holder) in the microscope. The former is the most common way. Most biprisms can be
moved in two horizontal directions (e.g. x: ±1.5 mm, y: ±1.0 mm ) and with an 180° in-plane rotation. The optimum position for the biprism is normally placed in the first image plane or sometimes in the second image plane. The position of the biprism changes the angle of the interfering electron beams and thus, for the same biprism potential, one sees fringes of different periodicity.
The interference width in the hologram, W, can be given by ,
α -- The deflection angle of the ray due to the action of
a -- The distance of the back focal
plane of the objective lens to the biprism,
R -- The radius of the biprism wire.
As shown in Figure 4308a, the two parts of the electron beam originate equivalently from two sources, S1 and S2, induced by the biprism, as in the standard Young’s experiment. The highly coherent beam induces in-phase over the interference region. A large potential must be applied to the biprism in order to move the two virtual sources further apart and to increase the width of the interference region (W). According to Equation 4298b in Young’s experiment, the fringe spacing will decrease as the biprism potential increases. Following the expression format of Figure 4308a, the fringe spacing can be re-written by,
ΔS = λ[(a+b)/(2αa)] ----------------------- [4308f]
Note that the smallest detectable phase difference (δϕmin) between adjacent pixels at an expected signal/noise ratio (SNR) is given by, 
N -- The number of electrons per resolved pixels,
DQE -- The detection quantum efficiency of the CCD camera.
Figure 4308b shows the schematic illustration of the effect of limited spatial coherence on the off-axis electron holograms. The electrons, emitted from two different locations in distance Δu in the back focal plane, form two inference patterns with a lateral shift of Δx. Both interference patterns superimpose incoherently, resulting in a reduced interference fringe contrast .
Figure 4308b. Schematic illustration of the effect of limited spatial coherence on the off-axis electron holograms.
is the spatial frequency
, while x
has units of length.
In order to obtain final off-axis holography images, the second step of electron holography analysis (reconstruction of the recorded off-axis holograms) needs to be performed (see page2560).
Research and engineering applications of off-axis electron holography can be split into three main categories:
i) Correct microscope aberration to achieve high spatial resolution;
ii) Measure electric fields in materials at high spatial resolution;
iii) Measure magnetic fields at high spatial resolution.
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