The electrons in the beam in EMs undergo different phase shifts when propagating through the objective lens because every electron source has a finite energy width. The maximum phase shift due to the energy width of the electron source is given by,
Cc -- Chromatic aberration coefficient
ΔE -- Energy width of the electron source
E0 -- Accelerating voltage of the electron beam
λ -- Wavelength of the electron beam
k -- Spatial frequency
Conventionally, the phase shift (phase distortion function) due to the objective lens can be combined into a single phase factor χ, given by,
Cs -- The spherical aberration coefficient, defining the quality of objective lens,
Δf -- The defocus value,
|g| -- Also the spatial frequency, the same as k above,
α -- The convergence semi-angle.
Considering the non-constant wavelength of an incident electron beam and the astigmatism, the phase shift function can be described by,
The last and the second terms describe the effects of the non-constant wavelength and astigmatism, respectively.
Note that for an ideal lens, the phase shift induced by objective lens, χ(α), is equal to 0.