== Linear Imaging Theory vs. Non-Linear Imaging Theory ==
=== 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, there are no dynamical effects, 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> One advantage of linear imaging theory is that 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>
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 ===
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In realpractically 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 incorporateproduce second order diffraction intensityintensities. 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 = |pmid = |access-date = }}</ref><ref>This page was prepared in part for Northwestern University class MSE 465, taught by Professor Laurie Marks. </ref>
