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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 [[cryo-electron microscopy]]. Its equivalent in light-based optics is the [[optical transfer function]].
==Phase
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 (waves)|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.
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Detectors are only able to directly measure the amplitude, not the phase. However, with the correct microscope parameters, the [[Interference (wave propagation)|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
[[File:TEM Ray Diagram with Phase Contrast Transfer Function.pdf|thumb|TEM Ray Diagram with Phase Contrast Transfer Function]]
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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
Following Wade's notation,<ref name=":0" /> the exit wavefunction expression is represented by:
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:<math>\tau(r,z) = \tau_o[1 + i\phi(r)]</math>
=== The
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:<br />
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== Examples of the
The contrast transfer function determines how much phase signal gets transmitted to the real space wavefunction in the image plane. As the modulus squared of the real space wavefunction gives the image signal, the contrast transfer function limits how much information can ultimately be translated into an image. The form of the contrast transfer
function determines the quality of real space image formation in the TEM.
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* Accordingly, past the point resolution of the microscope the phase information is not directly interpretable, and must be modeled via computer simulation
=== Scherzer
The defocus value (<math display="inline">z</math>) can be used to counteract the spherical aberration to allow for greater phase contrast. This analysis was developed by Scherzer, and is called the Scherzer defocus.<ref>{{Cite journal|url = http://scitation.aip.org/content/aip/journal/jap/20/1/10.1063/1.1698233 | doi = 10.1063/1.1698233 |title = The theoretical resolution limit of the electron microscope|last = Scherzer|first = |date = 1949|journal = Journal of Applied Physics|pmid = |access-date = |volume=20 |pages=20|bibcode = 1949JAP....20...20S }}</ref>
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The figure in the following section shows the CTF function for a CM300 Microscope at the Scherzer Defocus. Compared to the CTF Function showed above, there is a larger window, also known as a ''passband,'' of spatial frequencies with high transmittance. This allows more phase signal to pass through to the image plane.
=== Envelope
[[File:CTF Modified by Spatial and Temporal Envelope Functions.pdf|thumb|CTF Function of a CM300 Microscope damped by temporal and spatial envelope functions.]]
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Various software have been developed to model both the Contrast Transfer Function and Envelope Function for particular microscopes, and particular imaging parameters.<ref>{{cite web|title=CTF Simulation|url=http://jiang.bio.purdue.edu/software/ctf/ctfapplet.html|website=Wen Jiang Group|accessdate=27 April 2017}}</ref><ref>{{cite web|last1=Sidorov|first1=Max|title=Home of the ctfExplorer|url=http://www.maxsidorov.com/ctfexplorer/|accessdate=27 April 2017}}</ref>
== Linear
=== Linear
<br />
The previous description of the contrast transfer function depends on ''linear imaging theory''. Linear imaging theory assumes that the transmitted beam is dominant, there is only weak phase scattering by the sample, and that the sample is extremely thin. Linear imaging theory corresponds to all of the scattering, or diffraction, being [[Diffraction formalism|kinematical]] in nature. Few of these assumptions hold with real samples. In fact, even a single layer of Uranium atoms does not meet the Weak Phase Object Approximation.<ref>{{Cite book|title = Transmission Electron Microscopy:|last = Williams, Carter|first = |publisher = Springer|year = 2009|isbn = 978-0-387-76500-6|___location = |pages = }}</ref>
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Linear imaging theory is still used, however, because it has some computational advantages. In Linear imaging theory, the Fourier coefficients for the image plane wavefunction are separable. This greatly reduces computational complexity, allowing for faster computer simulations of HRTEM images.<ref>[http://www.numis.northwestern.edu/465/index.shtml Notes] prepared by Professor Laurie Marks at Northwestern University.</ref>
=== Non-
In practically all crystalline samples, the specimens will be strong scatterers, and will include multiple scattering events. This corresponds to [[Dynamical theory of diffraction|dynamical diffraction]]. In order to account for these effects, ''non-linear imaging theory'' is required. With crystalline samples, diffracted beams will not only interfere with the transmitted beam, but will also interfere with each other. This will produce second order diffraction intensities. Non-linear imaging theory is required to model these additional interference effects.<ref>{{Cite journal|url = http://www.sciencedirect.com/science/article/pii/0304399188902306|title = Contrast Transfer Theory for Non-Linear Imaging|last = Bonevich, Marks|first = |date = May 24, 1988|journal = Ultramicroscopy|doi = 10.1016/0304-3991(88)90230-6|pmid = |access-date =|volume=26|pages=313–319}}</ref><ref>This page was prepared in part for Northwestern University class MSE 465, taught by Professor Laurie Marks.</ref>
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