Practical Electron Microscopy and Database

The plane wave of electron beam propagating in z-direction (optical axis) can be given by,

          ---------------------- [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,

          (for x < 0 in Figure 4202) -------- [4202b]

          (for x > 0 in Figure 4202) -------- [4202c]

where
          k_⊥ ≈ kβ/2
          k_z ≈ k

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 and of the two partial waves with the unit vector e_x in x direction, forming the intensity (I) in a format of cosinoidal interference pattern,

          -------- [4202d]

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

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