While all the electron wavefunctions we have simulated are complex-valued, our physical detectors only allow us to capture the intensity (squared amplitude, $I(\bm{r}) = \left|\psi(\bm{r})\right|^2$) of the exit waves, as we saw in Detectors and Frozen Phonons. This loss of phase information is a well-known limitation of electron microscopy (and all other diffraction and microscopy fields), known as the Phase problem. To overcome this, one of the most powerful operating modes for HRTEM is Phase-contrast imaging (PCI) where imaging optics or detector configurations are used to convert phase modulations of the electron beam into intensity variations Springer International Publishing, 2016.

Source:Phase Contrast Imaging

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Imaging with Defocus¶

The simplest method to produce phase contrast in HRTEM imaging is to apply under- or over-focus to the electron wave after it interacts with the sample, shown in Figure 9.1. The protein sample shown here is apoferritin, which produces very weak diffraction of the electron beam. To produce usable contrast from defocus, we must either apply a large defocus or increase the electron fluence, colloquially referred to as the electron dose Egerton, 2021.

Defocusing an electron wavefunction $\psi_0(\bm{r})$ by a distance $\Delta f$ to produce the output wave $\psi(\bm{r})$ can be modeled mathematically using the expression

The electron wavefunction after interacting with a weakly-scattering sample can be approximated as

where the $\phi(\bm{r})$ is the phase shift imparted by the sample, and it is considered to be a weak phase object Vulović et al., 2014. Combining equations (9.1) and (9.2), we can derive the measured intensity for a weak phase object to be

Imaging with Phase Plates¶

An alternative PCI method for plane wave TEM is to use a post-specimen phase plate which advances the phase of the unscattered zero beam with respect to the scattered electrons, or vice versa. These phase plates have various designs, including a Zernike phase plate Zernike, 1942, Boersch phase plate Boersch, 1947, Volta phase plate Danev et al., 2014, or the recently developed laser phase plates Schwartz et al., 2017. An ideal phase plate can be modeled in diffraction space using the expression

where $\delta(\bm{k})$ is the Dirac delta function. The sample intensity for a weak phase object is given by the expression

Figure 9.1 shows how Zernike phase plate PCI compares to defocusing the sample. We note however that practical implementations of phase plates remaining challenging, and typically do not perform as well as the ideal case shown aboveNagayama, 2011.