Practical Electron Microscopy and Database

In electron-optical systems, (as well as light-optical systems), lens can work as an aperture because it can constraint the incident wave front. This is more easily understood for light-optical systems because the aperture effect is due to the finite surface area of the optical lens;

Assuming the incident wave has a complex wave function just before the lens, given by,

              --------------------------- [4159a]

After the incident wave crosses the lens, the wave function becomes,

              --------------------------- [4159b]

So that the aperture function of the lens is given by,

              ------------------- [4159c]

                     ------------------ [4159d]

where,
          D -- The diameter of the lens
          r -- The distance from the incident point to the optic axis of the lens (r = (x₂+y₂)^1/2)

The lens behavior constraining the wave plane to r < D/2 is so-called aperture function of the lens. For electron-optical system, the absorption of the lens is zero so that a(x,y) = A₂/A₁ ≈ 1. Therefore, the lens function becomes pure phase function.