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== Contrast transfer theory ==
[[File:TEM Ray Diagram with Phase Contrast Transfer Function.pdf|thumb|TEM Ray Diagram with Phase Contrast Transfer Function]]
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==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.''
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=== 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:<br />
:<math>I(\theta) = \delta(\theta) + \Phi K(\theta)</math>
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[[File:Unmodified CTF.pdf|thumb|CTF Function prepared via web applet created by Jiang and Chiu, available at https://ctfsimulation.streamlit.app/]]
This is an example contrast transfer function. There are a number of things to note:
* The function exists in the spatial frequency ___domain, or k-space
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=== Scherzer defocus ===
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| doi = 10.1063/1.1698233 |title = The theoretical resolution limit of the electron microscope|last = Scherzer|date = 1949|journal = Journal of Applied Physics|volume=20 |issue = 1|pages=20–29|bibcode = 1949JAP....20...20S }}</ref>
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== Linear imaging theory vs. non-linear imaging theory ==
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 shift by the sample. In many cases, this precondition is not fulfilled. In order to account for these effects, ''non-linear imaging theory'' is required. With strongly scattering samples, diffracted electrons 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|title = Contrast Transfer Theory for Non-Linear Imaging|last = Bonevich, Marks|date = May 24, 1988|journal = Ultramicroscopy|doi = 10.1016/0304-3991(88)90230-6|volume=26|issue = 3|pages=313–319}}</ref><ref>This page was prepared in part for Northwestern University class MSE 465, taught by Professor [[Laurence D. Marks|Laurie Marks]].</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 [[Laurence D. Marks|Laurie Marks]] at Northwestern University.</ref>
== See also ==
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