Contrast transfer function

The contrast transfer function (CTF) of the electron microscope specifies the relative contrast of features in the image for all spatial frequency ranges and depends both on the spherical aberration and the level of focus of the objective lens.

In the image below it is visualized how the projection image can differ from the sample depending on the contrast transfer function:

[img[1][EmbeddedImages]]

$T(H)=\sin(\pi *Cs\lambda ^{3}*H^{4}/2+\pi \delta f*\lambda H^{2})$

Cs: Spherical aberration constant. $\delta f$: defocus value. $\lambda$: the electron wave length

The exit wave passes through the imaging system of the microscope where it undergoes further phase change and interferes as the image wave in the imaging plane. A recorded image is NOT a direct representation of the samples crystallographic structure. For instance, high intensity might or might not indicate the presence of an atom column in that precise location. The relationship between the exit wave and the image wave is a highly nonlinear one and is a function of the aberrations of the microscope. It is described by the contrast transfer function.

[img[2][EmbeddedImages]]

H denotes the spatial frequency.

T(H)<0 implies “positive” constrast: atom columns appear dark (in the print, not the negative!). T(H)>0 implies “negative” contrast: atom columns appear bright. • T(H) = 0 implies no transfer of the respective spatial frequency at all!

Example: hypothetical crystal with four different sets of planes parallel to the viewing direction – plane spacing: d1 > d2 > d3 > d4 – corresponding spatial frequencies: 1/d1 < 1/d2 < 1/d3 < 1/d4. – the planes with spacing d1 appear with positive contrast – the planes with spacing d2 appear with negative contrast – the planes with spacing d3 do not appear at all – it is difficult to predict the contrast of the planes with spacing d4. We can avoid these problems by introducing an objective aperture.