The contrast transfer function (CTF) mathematically describes how aberrations in a
transmission electron microscope (TEM) modify the image of a sample.
[Spence, John C. H. (1988 2nd ed) ''Experimental high-resolution electron microscopy'' (Oxford U. Press, NY) .][Ludwig Reimer (1997 4th ed) ''Transmission electron microscopy: Physics of image formation and microanalysis'' (Springer, Berlin]
preview
[Earl J. Kirkland (1998) ''Advanced computing in electron microscopy'' (Plenum Press, NY).] This contrast transfer function (CTF) sets the resolution of
high-resolution transmission electron microscopy
High-resolution transmission electron microscopy is an imaging mode of specialized transmission electron microscopes that allows for direct imaging of the atomic structure of samples. It is a powerful tool to study properties of materials on the a ...
(HRTEM), also known as phase contrast TEM.
By considering the recorded image as a CTF-degraded true object, describing the CTF allows the true object to be
reverse-engineered
Reverse engineering (also known as backwards engineering or back engineering) is a process or method through which one attempts to understand through deductive reasoning how a previously made device, process, system, or piece of software accompli ...
. This is typically denoted CTF-correction, and is vital to obtain high resolution structures in three-dimensional electron microscopy, especially
electron cryo-microscopy. Its equivalent in light-based optics is the
optical transfer function
The optical transfer function (OTF) of an optical system such as a camera, microscope, human eye, or projector specifies how different spatial frequencies are captured or transmitted. It is used by optical engineers to describe how the optics pr ...
.
Phase contrast in HRTEM
The contrast in HRTEM comes from interference in the image plane between the phases of scattered
electron waves with the phase of the transmitted electron wave. When an electron wave passes through a sample in the TEM, complex interactions occur. Above the sample, the electron wave can be approximated as a plane wave. As the electron wave, or
wavefunction, passes through the sample, both the
phase and the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of ampl ...
of the electron beam is altered. The resultant scattered and transmitted electron beam is then focused by an objective lens, and imaged by a detector in the image plane.
Detectors are only able to directly measure the amplitude, not the phase. However, with the correct microscope parameters, the
phase interference can be indirectly measured via the intensity in the image plane. Electrons interact very strongly with
crystalline
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
solids. As a result, the phase changes due to very small features, down to the atomic scale, can be recorded via HRTEM.
Contrast transfer theory
Contrast transfer theory provides a quantitative method to translate the exit wavefunction to a final image. Part of the analysis is based on
Fourier transforms of the electron beam wavefunction. When an electron wavefunction passes through a lens, the wavefunction goes through a Fourier transform. This is a concept from
Fourier optics.
Contrast transfer theory consists of four main operations:
# Take the Fourier transform of the exit wave to obtain the wave amplitude in back focal plane of objective lens
# Modify the wavefunction in reciprocal space by a phase factor, also known as the ''Phase Contrast Transfer Function'', to account for aberrations
# Inverse Fourier transform the modified wavefunction to obtain the wavefunction in the image plane
# Find the square modulus of the wavefunction in the image plane to find the image intensity (this is the signal that is recorded on a detector, and creates an image)
Mathematical form
If we incorporate some assumptions about our sample, then an analytical expression can be found for both phase contrast and the phase contrast transfer function. As discussed earlier, when the electron wave passes through a sample, the electron beam interacts with the sample via scattering, and experiences a phase shift. This is represented by the electron wavefunction exiting from the bottom of the sample. This expression assumes that the scattering causes a phase shift (and no amplitude shift). This is called the ''Phase Object Approximation.''
The exit wavefunction
Following Wade's notation,
the exit wavefunction expression is represented by:
: