Contrast transfer function: Difference between revisions

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[[ImageFile: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.

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The contrast transfer function can also be given in terms of spatial frequencies, or reciprocal space. With the relationship <math display="inline">\theta =\lambda k</math>, the phase contrast transfer function becomes:

:<math>K(k) = \sin[(2\pi\lambda) )W(k)]</math><br /> <math>W(k) = -z\lambda k^2/2 + C_s\lambda^3 k^4/4</math>

<math>z</math> = the defocus of the objective lens (using the convention that underfocus is positive and overfocus is negative), <math>\lambda</math> = the relativistic wavelength of the electron wave, <math>C_s</math> = The [[spherical aberration]] of the objective lens, <math>k</math> = the spatial frequency (units of m⁻¹)