Contrast transfer function

Did You Know?

• Contrast transfer theory provides a quantitative method to translate the exit wavefunction to a final image, based on Fourier transforms of the electron beam wavefunction.
• Contrast transfer theory consists of four main operations: Fourier transform of the exit wave, modification by a phase factor, inverse Fourier transform, and taking the square modulus to find image intensity.
• The equivalent of the contrast transfer function in light-based optics is the optical transfer function.

Phase contrast in HRTEM

Contrast in HRTEM arises from interference in the image plane between the phases of scattered electron waves and the phase of the transmitted electron wave. Above the sample, the electron wave approximates a plane wave. As it passes through the sample, both phase and amplitude are altered. The resultant beam is focused by an objective lens and imaged by a detector. Detectors measure only amplitude, but with correct microscope parameters, phase interference can be indirectly measured via intensity. Electrons interact strongly with crystalline solids, allowing phase changes from atomic-scale features to be recorded.

Mathematical form

Under the Phase Object Approximation, the exit wavefunction assumes scattering causes only a phase shift, with no amplitude shift. Following Wade's notation, the exit wavefunction τ depends on coordinates in the sample plane and perpendicular to it, with the incident wavefunction, electron wavelength set by accelerating voltage, and the sample's effective potential. The phase shift is represented by an expression involving these quantities. If the sample is very thin and a weak scatterer, the phase shift is much less than 1, and the wavefunction can be approximated by a linear Taylor expansion, known as the Weak Phase Object Approximation. The exit wavefunction then takes a simplified form. Passing through the objective lens incurs a Fourier transform and phase shift. The wavefunction on the back focal plane includes a delta function for the transmitted wave, the Fourier transform of the phase, and the phase shift from aberrations, called the Contrast Transfer Function. This function depends on the relativistic wavelength, spherical aberration, and scattering angle. In terms of spatial frequencies, the phase contrast transfer function becomes a function of defocus, relativistic wavelength, spherical aberration, and spatial frequency.

Examples

The contrast transfer function determines how much phase signal is transmitted to the real space wavefunction in the image plane. Since the image signal is the modulus squared of that wavefunction, the CTF limits the information translated into an image. The function exists in the spatial frequency domain (k-space). Where the function equals zero, no phase signal is incorporated into the real space image. The first zero crossing is called the point resolution. To maximize phase signal, imaging conditions that push the point resolution to higher spatial frequencies are generally better. When the function is negative, positive phase contrast results, giving a bright background with dark atomic features. Each zero crossing inverts contrast. Past the point resolution, phase information is not directly interpretable and must be modeled by computer simulation. Defocus can counteract spherical aberration to allow greater phase contrast, an analysis developed by Scherzer and called the Scherzer defocus. The envelope function represents additional aberrations that damp the CTF, suppressing high spatial frequencies. The effective CTF is obtained by multiplying the CTF by envelope terms for temporal and spatial aberrations.