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== Linear imaging theory vs. non-linear imaging theory ==
=== Linear vs. nonlinear 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 scatteringshift by the sample,. andIn thatmany thecases, samplethis precondition is extremelynot thinfulilled. LinearIn order to account for these effects, ''non-linear imaging theory'' correspondsis torequired. allWith of thestrongly scattering samples, ordiffracted diffraction,electrons beingwill [[Diffractionnot formalism|kinematical]]only ininterfere nature.with Fewthe oftransmitted thesebeam, assumptionsbut holdwill also interfere with realeach samplesother. InThis fact,will evenproduce asecond singleorder layerdiffraction ofintensities. UraniumNon-linear atomsimaging doestheory notis meetrequired theto Weakmodel these Phaseadditional Objectinterference Approximationeffects.<ref>{{Cite bookjournal|title = TransmissionContrast ElectronTransfer MicroscopyTheory for Non-Linear Imaging|last = WilliamsBonevich, CarterMarks|publisherdate = SpringerMay 24, 1988|yearjournal = 2009Ultramicroscopy|isbndoi = 97810.1016/0304-0-387-765003991(88)90230-6|___locationpmid = |pagesvolume=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 imaging theory has nothing to do with [[Diffraction formalism|kinematical diffraction]] or [[Dynamical theory of diffraction|dynamical diffraction]], respectively.
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|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>
== See also ==
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