Contrast transfer function: Difference between revisions

Revision as of 19:34, 12 June 2015

Revision as of 05:51, 13 June 2015

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==Phase Contrast in HRTEM==

The contrast in '''HRTEM''' comes from interference in the image plane between the phases of scattered electron waves with the phase of the transmitted electron wave.  When an electron wave passes through a sample in the TEM, complex interactions occur. Above the sample, the electron wave can be approximated as a plane wave. As the electron wave, or wavefunction, passes through the sample, both the phase and the amplitude of the electron beam is altered.  The resultant scattered and transmitted electron beam is then focused by an objective lens, and imaged by a detector in the image plane.

Detectors are only able to directly measure the amplitude, not the phase.  However, with the correct microscope parameters, the phase interference can be indirectly measured via the intensity in the image plane. Electrons interact very strongly with crystalline solids. As a result, the phase changes due to very small features, down to the atomic scale, can be recorded via HRTEM.

The contrast in '''HRTEM''' comes from interference in the image plane between the phases of scattered electron waves with the phase of the transmitted electron wave.  When an electron wave passes through a sample in the TEM, complex interactions occur. Above the sample, the electron wave can be approximated as a plane wave. As the electron wave, or wavefunction, passes through the sample, both the phase and the amplitude of the electron beam is altered.  The resultant scattered and transmitted electron beam is then focused by an objective lens, and imaged by a detector in the image plane.

== Contrast Transfer Theory ==

Detectors are only able to directly measure the amplitude, not the phase.  However, with the correct microscope parameters, the phase interference can be indirectly measured via the intensity in the image plane. Electrons interact very strongly with crystalline solids. As a result, the phase changes due to very small features, down to the atomic scale, can be recorded via HRTEM.

== Contrast Transfer Theory ==

[[File:TEM Ray Diagram with Phase Contrast Transfer Function.pdf|thumb|TEM Ray Diagram with Phase Contrast Transfer Function]]

'''Contrast Transfer Theory''' provides a quantitative method to translate the exit wavefunction to a final image. Part of the analysis is based on Fourier transforms of the electron beam wavefunction.  When an electron wavefunction passes through a lens, the wavefunction goes through a [[Fourier transform]]. This is a concept from [[Fourier optics]].

Contrast Transfer Theory consists of four main operations:<ref name=":0" />

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==Mathematical form==

If we incorporate some assumptions about our sample, then an analytical expression can be found for both '''phase contrast''' and the phase '''contrast transfer function'''.  As discussed earlier, when the electron wave passes through a sample, the electron beam interacts with the sample via scattering, and experiences a phase shift.  This is represented by the electron wavefunction exiting from the bottom of the sample. This expression assumes that the scattering causes a phase shift (and no amplitude shift).  This is called the '''Phase Object Approximation.'''

=== The Exit Wavefunction ===

Following Wade's notation,<ref name=":0" />, the exit wavefunction expression is represented by:

:<math>\tau(r,z) = \tau_o exp[-i\pi\lambda\int dz'U(r,z')]</math> <br />

:<math>\tau_o = \tau(r,0)</math> <br />

:<math>U(r,z) = 2mV(r,z)/h^2</math>

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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|accessdate = 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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=== The Phase Contrast Transfer Function ===

Passing through the objective lens incurs a Fourier transform and phase shift.  As such, the wavefunction on the back focal plane of the objective lens can be represented by:<br />

:<math>I(\theta) = \delta(\theta) + \Phi K(\theta)</math>

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

<math>\lambda</math> = the relativistic wavelength of the electron wave, <math>C_s</math> = The [[spherical aberration]] of the objective lens

<br />

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:<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</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^-1−1)

===Spherical aberration===

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== Examples of the Contrast Transfer Function ==

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]]

This is an example contrast transfer function.  There are a number of things to note:

This is an example contrast transfer function. There are a number of things to note:

* The function exists in the spatial frequency ___domain, or k-space

* Whenever the function is equal to zero, that means there is no transmittance, or no phase signal is incorporated into the real space image

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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|url = http://scitation.aip.org/content/aip/journal/jap/20/1/10.1063/1.1698233|title = The theoretical resolution limit of the electron microscope|last = Scherzer|first = |date = 1949|journal = Journal of Applied Physics|doi = |pmid = |access-date = }}</ref>

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

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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.]]

  <nowiki> </nowiki>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>

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|accessdate = 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. Becuase 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.

Various [http://jiang.bio.purdue.edu/software/ctf/ctfapplet.html software] [http://www.maxsidorov.com/ctfexplorer/ packages] have been developed to model both the Contrast Transfer Function and Envelope Function for particular microscopes, and particular imaging parameters.

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=== Linear Imaging Theory ===

<br />

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, 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>

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>

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=== Non-Linear Imaging Theory ===

<br />

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|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>

== See also ==

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* [http://www.youtube.com/watch?v=I3_4HF1ZeIQ Talk on the CTF by Henning Stahlberg]

* [http://em-outreach.ucsd.edu/web-course/ref2.html CTF reading list]

* [http://www.maxsidorov.com/ctfexplorer/index.htm Interactive CTF Modeling]

== References ==