Contrast transfer function: Difference between revisions

Revision as of 08:26, 27 October 2020

Revision as of 12:43, 14 December 2020

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[[File: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|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 = |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 [[electron cryo-microscopy]]. Its equivalent in light-based optics is the [[optical transfer function]].

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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> is the wavelength of the electron beam,<ref>{{cite web|title=DeBroglie Wavelength|url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/debrog2.html#c1|website=HyperPhysics|publisher=Georgia State University|accessdateaccess-date=27 April 2017}}</ref> which is set by the accelerating voltage. <math>U</math> is the effective potential of the sample, which depends on the atomic potentials within the crystal, represented by <math>V</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|accessdateaccess-date = 2015-06-12}}</ref> This approximation is called the ''Weak Phase Object Approximation.''

The exit wavefunction can then be expressed as:

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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|pmid = |volume=20 |issue = 1|pages=20–29|bibcode = 1949JAP....20...20S }}</ref>

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

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<math>K_{eff}(k) = E_tE_s(\sin[(2\pi/\lambda)W(k)]</math>

Examples of temporal aberrations include chromatic aberrations, energy spread, focal spread, instabilities in the high voltage source, and instabilities in the objective lens current. An example of a spatial aberration includes the finite incident beam convergence.<ref>{{Cite web|title = Envelope Functions|url = http://www.maxsidorov.com/ctfexplorer/webhelp/envelope_functions.htm|website = www.maxsidorov.com|accessdateaccess-date = 2015-06-12}}</ref>

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<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 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|accessdateaccess-date=27 April 2017}}</ref><ref>{{cite web|last1=Sidorov|first1=Max|title=Home of the ctfExplorer|url=http://www.maxsidorov.com/ctfexplorer/|accessdateaccess-date=27 April 2017}}</ref>

== 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 fulilled. 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|pmid = |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>

Contrary to a widespread assumption, the linear/nonlinear imaging theory has nothing to do with [[Diffraction formalism|kinematical diffraction]] or [[Dynamical theory of diffraction|dynamical diffraction]], respectively.