Plane Wave and Interference Pattern in Electron Holography
- Practical Electron Microscopy and Database -
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The plane wave of electron beam propagating in z-direction (optical axis) can be given by,

          Plane Wave of Electron Beam ---------------------- [4202a]

where,
         a -- Amplitude
         k -- Wavenumber (=1/λ)
         λ -- Wavelength
         ω -- Frequency

The vector r = (x, y) defines a position in a wavefront perpendicular to the electron propagation direction z. In off-axis electron holography as shown in Figure 4202, the plane wave of the electron beam is split to two partial waves which are deflected by a very small angle towards each other,

          two partial waves in off-axis electron holography(for x < 0 in Figure 4202) -------- [4202b]

          two partial waves in off-axis electron holography(for x > 0 in Figure 4202) -------- [4202c]

where
          k ≈ kβ/2
          kz ≈ k

Schematic diagram of off-axis electron holography in TEMs

Figure 4202. Schematic diagram of off-axis electron holography in TEMs.

The two tilted waves are laterally shifted to the right (for the left part) and to the left (for the right part) by a width of W, respectively, and thus are superimposed in the hologram (W). In the point r of the detector the points superimposed in the hologram (W).and superimposed in the hologram (W). of the two partial waves with the unit vector ex in x direction, forming the intensity (I) in a format of cosinoidal interference pattern,

          cosinoidal term produces the fringes in the measurements of off-axis electron holography -------- [4202d]

where,
          qc - Spatial frequency (=kβ)
          β -- The angle of the superposition

The cosinoidal term produces the fringes in the measurements of off-axis electron holography.

 

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