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[[Image:Contrast transfer function.jpg|thumb|Power spectrum (Fourier transform) of a typical electron micrograph. The effect of the contrast transfer function can be seen in the alternating light and dark rings (Thon rings), which show the relation between contrast and spatial frequency. ]]
The '''contrast transfer function''' (CTF) mathematically describes how aberrations in a
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 Contrast in HRTEM==
The contrast in
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 Transfer Theory ==
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[[File:TEM Ray Diagram with Phase Contrast Transfer Function.pdf|thumb|TEM Ray Diagram with Phase Contrast Transfer Function]]
Contrast
# 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
# 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)
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==Mathematical form==
If we incorporate some assumptions about our sample, then an analytical expression can be found for both
=== The Exit Wavefunction ===
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:<math>U(r,z) = 2mV(r,z)/h^2</math>
Where the exit wavefunction τ is a function of both <math>r</math> in the plane of the sample, and <math>z</math> perpendicular to the plane of the sample. <math>\tau_o</math> represents the wavefunction incident on the top of the sample. <math>\lambda</math>
Within the exit wavefunction, the phase shift is represented by:
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:<math>\phi(r) = \pi\lambda \int dz' U(r,z')</math>
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]].<ref>{{Cite web|title = Weak-phase-objects (WPO) in TEM observations - Practical Electron Microscopy and Database - An Online Book - EELS EDS TEM SEM|url = http://www.globalsino.com/EM/page4173.html|website = www.globalsino.com|accessdate = 2015-06-12}}</ref> This approximation is called the
The exit wavefunction can then be expressed as:
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<math>\Phi</math> = the Fourier transform of the wavefunction's phase
<math>K(\theta)</math> = the phase shift incurred by the microscope's aberrations, also known as the
:<math>K(\theta) = \sin[(2\pi/\lambda)W(\theta)]</math> <br /><math>W(\theta) = -z\theta^2/2 + C_s\theta^4/4</math>
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===Spherical aberration===
[[Spherical aberration]] is a blurring effect arising when a lens is not able to converge incoming rays at higher angles of incidence to the focus point, but rather focuses them to a point closer to the lens. This will have the effect of spreading an imaged point (which is ideally imaged as a single point in the [[gaussian]] image plane) out over a finite size disc in the image plane. Giving the measure of aberration in a plane normal to the optical axis is called a transversal aberration. The size (radius) of the aberration disc in this plane can be shown to be proportional to the cube of the incident angle (θ) under the small-angle approximation, and that the explicit form in this case is
:<math>
r_s = C_s\cdot\theta^3\cdot M
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===Defocus===
As opposed to the spherical aberration, we will proceed by estimating the deviation of a defocused ray from the ideal by stating the longitudal aberration; a measure of how much a ray deviates from the focal point along the optical axis. Denoting this distance <math>\Delta b</math>, it is possible to show that the difference <math>\alpha_f</math> in refracted angle between rays originating from a focused and defocused object, can be related to the refracted angle as<br />
:<math>
\sqrt{R^2+b^2}\cdot\sin(\alpha_f)=\Delta b \cdot\sin(\theta' -\alpha_f)
</math>
where <math>R</math> and <math>b</math> are defined in the same way as they were for spherical aberration. Assuming that <math>\alpha_f<<\theta'</math> (or equivalently that <math>|b\cdot\sin(\alpha_f)|<<|R|</math> ), we can show that <br />
:<math>
\sin(\alpha_f)\approx\frac{\Delta b \sin(\theta')}{\sqrt{R^2 +b^2}} = \frac{\Delta b \cdot R}{R^2 +b^2}
</math>
Since we required <math>\alpha_f</math> to be small, and since <math>\theta</math> being small implies <math>R<<b</math>, we are given an approximation of <math>\alpha_f</math> as<br />
:<math>
\alpha_f\approx\frac{\Delta b\cdot R}{b^2}
</math>
From the [[thin-lens formula]] it can be shown that <math>\Delta b / b^2 \approx \Delta f / f^2</math>, yielding a final estimation of the difference in refracted angle between in-focus and off-focus rays as<br />
:<math>
\alpha_f\approx\frac{\Delta f\cdot R}{f^2}
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* The function exists in the spatial frequency ___domain, or k-space
* Whenever the function is equal to zero, that means there is no transmittance, or no phase signal is incorporated into the real space image
* The first time the function crosses the x-axis is called the
* To maximize phase signal, it is generally better to use imaging conditions that push the point resolution to higher spatial frequencies
* When the function is negative, that represents positive phase contrast, leading to a bright background, with dark atomic features
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The variables are the same as from the mathematical treatment section, with <math>z_s</math> setting the specific Scherzer defocus, <math>C_s</math> as the spherical aberration, and λ as the relativistic wavelength for the electron wave.
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
=== Envelope Function ===
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<br />
As shown in the figure, the most restrictive envelope term will dominate in damping the contrast transfer function. In this particular example, the temporal envelope term is the most restrictive. Because the envelope terms damp more strongly at higher spatial frequencies, there comes a point where no more phase signal can pass through. This is called the
<br /> Modeling the envelope function can give insight into both TEM instrument design, and imaging parameters. By modeling the different aberrations via envelope terms, it is possible to see which aberrations are most limiting the phase signal.
Various
== Linear Imaging Theory vs. Non-Linear Imaging Theory ==
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=== Linear Imaging Theory ===
<br />
The previous description of the contrast transfer function depends on
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-Linear Imaging Theory ===
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,
== See also ==
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* [[Optical transfer function]]
* [[Point spread function]]
* [http://www.wadsworth.org/spider_doc/spider/docs/techs/ctf/ctf.html Contrast transfer function (CTF) correction]▼
* [[Airy disk]], different but similar phenomena in light
* [https://www.youtube.com/watch?v=I3_4HF1ZeIQ Talk on the CTF by Henning Stahlberg]▼
* [http://em-outreach.ucsd.edu/web-course/ref2.html CTF reading list]▼
* [http://www.maxsidorov.com/ctfexplorer/index.htm Interactive CTF Modeling]▼
== References ==
<references/>
==External links==
▲* [http://www.wadsworth.org/spider_doc/spider/docs/techs/ctf/ctf.html Contrast transfer function (CTF) correction]
▲* [https://www.youtube.com/watch?v=I3_4HF1ZeIQ Talk on the CTF by Henning Stahlberg]
▲* [http://em-outreach.ucsd.edu/web-course/ref2.html CTF reading list]
▲* [http://www.maxsidorov.com/ctfexplorer/index.htm Interactive CTF Modeling]
[[Category:Microscopes]]
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