Contrast transfer function
The contrast transfer function (CTF) mathematically describes how aberrations in a transmission electron microscope (TEM) modify the image of a sample.
Complex interactions occur when an electron wave passes through a sample in the TEM.
However, with the correct microscope parameters, the phase interference can be indirectly measured via the intensity in the image plane.
As a result, the phase changes due to very small features, down to the atomic scale, can be recorded via HRTEM.
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.
Within the exit wavefunction, the phase shift is represented by: 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.
The exit wavefunction can then be expressed as: Passing through the objective lens incurs a Fourier transform and phase shift.
= the phase shift incurred by the microscope's aberrations, also known as the Contrast Transfer Function:
= The spherical aberration of the objective lens The contrast transfer function can also be given in terms of spatial frequencies, or reciprocal space.
= the defocus of the objective lens (using the convention that underfocus is positive and overfocus is negative),
= the spatial frequency (units of m−1) Spherical aberration is a blurring effect arising when a lens is not able to converge incoming rays at higher angles of incidence to the focus point, but rather focuses them to a point closer to the lens.
The size (radius) of the aberration disc in this plane can be shown to be proportional to the cube of the incident angle (θ) under the small-angle approximation, and that the explicit form in this case is where
One can then go on to note that the difference in refracted angle between an ideal ray and one which suffers from spherical aberration, is where
is the radial distance from the optical axis to the point on the lens which the ray passed through.
By way of these assumptions we have also implicitly stated that the fraction itself can be considered small, and this results in the elimination of the
is again considered small, then meaning that an approximate expression for the difference in refracted angle between an ideal ray and one which suffers from spherical aberration, is given by As opposed to the spherical aberration, we will proceed by estimating the deviation of a defocused ray from the ideal by stating the longitudinal aberration; a measure of how much a ray deviates from the focal point along the optical axis.
, yielding a final estimation of the difference in refracted angle between in-focus and off-focus rays as The contrast transfer function determines how much phase signal gets transmitted to the real space wavefunction in the image plane.
The form of the contrast transfer function determines the quality of real space image formation in the TEM.
The figure in the following section shows the CTF function for a CM300 Microscope at the Scherzer Defocus.
Compared to the CTF Function showed above, there is a larger window, also known as a passband, of spatial frequencies with high transmittance.
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.
[8] As shown in the figure, the most restrictive envelope term will dominate in damping the contrast transfer function.
Because the envelope terms damp more strongly at higher spatial frequencies, there comes a point where no more phase signal can pass through.
Modeling the envelope function can give insight into both TEM instrument design, and imaging parameters.
[9][10] 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 shift by the sample.
Non-linear imaging theory is required to model these additional interference effects.
