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<math>\lambda</math> = the relativistic wavelength of the electron wave, <math>C_s</math> = The [[spherical aberration]] of the objective lens
<br />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: ▼
▲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) W(k)]</math><br /> <math>W(k) = -z\lambda k^2/2 + C_s\lambda^3 k^4/4</math>
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== Examples ==
The contrast transfer function determines how much phase signal gets transmitted to the real space wavefunction in the image plane. As the [[modulus squared]] of the real space wavefunction gives the image signal, the contrast transfer function limits how much information can ultimately be translated into an image. The form of the contrast transfer function determines the quality of real space image formation in the TEM.
[[File:Unmodified CTF.pdf|thumb|CTF Function prepared via web applet created by Jiang and Chiu, available at http://jiang.bio.purdue.edu/software/ctf/ctfapplet.html]]
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[[File:CTF Modified by Spatial and Temporal Envelope Functions.pdf|thumb|CTF Function of a CM300 Microscope damped by temporal and spatial envelope functions.]]
The envelope function represents the effect of additional aberrations that damp the contrast transfer function, and in turn the phase. The envelope terms comprising the envelope function tend to suppress high spatial frequencies. The exact form of the envelope functions can differ from source to source. Generally, they are applied by multiplying the Contrast Transfer Function by an envelope term Et representing temporal aberrations, and an envelope term Es representing spatial aberrations. This yields a modified, or effective Contrast Transfer Function:
<math>K_{eff}(k) = E_tE_s(\sin[(2\pi/\lambda)W(k)]</math>
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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|access-date = 2015-06-12}}</ref>
<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 ''Information Limit'' of the microscope, and is one measure of the resolution.▼
▲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 ''Information Limit'' of the microscope, and is one measure of the resolution.
<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.
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== Linear imaging theory vs. non-linear imaging theory ==
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