Contrast transfer function: Difference between revisions

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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 [[transmission electron microscope]] (TEM) modify the image of a sample.<ref name=":0">{{Cite journal|url = http://www.sciencedirect.com/science/article/pii/0304399192900118|title = A brief look at imaging and contrast transfer|last = Wade|first = R. H.|date = October 1992|journal = Ultramicroscopy|doi = 10.1016/0304-3991(92)90011-8|pmid = |access-date =|volume=46|issue = 1–4|pages=145–156}}</ref><ref name="Spence1982">Spence, John C. H. (1988 2nd ed) ''Experimental high-resolution electron microscopy'' (Oxford U. Press, NY) {{ISBN|0195054059}}.</ref><ref name="Reimer97">Ludwig Reimer (1997 4th ed) ''Transmission electron microscopy: Physics of image formation and microanalysis'' (Springer, Berlin) [https://books.google.com/books?id=3_84SkJXnYkC preview].</ref><ref name="Kirkland1998">Earl J. Kirkland (1998) ''Advanced computing in electron microscopy'' (Plenum Press, NY).</ref> 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 [[cryo-electron microscopy]]. Its equivalent in light-based optics is the [[optical transfer function]].

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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|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 |volumeissue =20 1|pages=2020–29|bibcode = 1949JAP....20...20S }}</ref>

<math>z_s = (C_s\lambda)^{1/2}</math>

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=== Linear imaging theory ===

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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>

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, ''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|issue = 3|pages=313–319}}</ref><ref>This page was prepared in part for Northwestern University class MSE 465, taught by Professor Laurie Marks.</ref>

== See also ==