The contrast transfer function (CTF) mathematically describes how aberrations in a transmission electron microscope (TEM) modify the image of a sample. This contrast transfer function (CTF) sets the resolution of high-resolution transmission electron microscopy (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. 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.
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 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 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:
Where the exit wavefunction τ is a function of both in the plane of the sample, and perpendicular to the plane of the sample. represents the wavefunction incident on the top of the sample. is the wavelength of the electron beam, which is set by the accelerating voltage. is the effective potential of the sample, which depends on the atomic potentials within the crystal, represented by .
Within the exit wavefunction, the phase shift is represented by:
This expression can be further simplified taken into account some more assumptions about the sample. If the sample is considered very thin, and a weak scatterer, so that the phase shift is << 1, then the wave function can be approximated by a linear Taylor polynomial expansion. This approximation is called the Weak Phase Object Approximation.
The exit wavefunction can then be expressed as:
The phase contrast transfer function
Passing through the objective lens incurs a Fourier transform and phase shift. As such, the wavefunction on the back focal plane of the objective lens can be represented by:
= the scattering angle between the transmitted electron wave and the scattered electron wave