# Phase contrast and information limit

Every lens image can be represented mathematically as a Fourier transformation from the front to the back focal plane. Thereby the spatial coordinates are transformed into their reciprocal values, commonly known as spatial frequencies. This means that small distances in the object correspond to large spatial frequencies and vice versa.

Whenever thin amorphous objects are imaged in a transmission electron microscope, the electrons undergo phase shifts imposed by the atoms of the specimen under investigation, whereas the intensity of the electron beam is not influenced. These phase shifts per se do not give rise to an image. Only the superposition of the undisturbed wave with the scattered wave leads to interference and thus to changes in intensity, which can be measured. The contrast will be optimized by a phase shift of ±π/2 between scattered and original wave. In light optical microscopy a phase plate introduces the required phase shift.

In an uncorrected electron microscope the intrinsic aberrations lead to a phase shift of the electron beam, the main part of which is caused by the spherical aberration. If a point (delta function) is imaged, the phase difference between scattered and unscattered electrons is given by

`\phi(k)=\frac{\pi}{2}\cdot(c_s\cdot\lambda^3\cdot{k^3}-2\cdot\Delta{f}\cdot\lambda\cdot{k^2})`

k denotes the spatial frequency, c_{s} the spherical aberration coefficient and Δf the defocus. The resulting wave function is

`ctf(k)=A_{n_\omicron}\cdot(cos\phi(k)-i\cdot{sin}\phi(k))`

The imaginary part of this function describes the phase transfer function for weak phase objects. Its amplitude A_{0} is given by the maximum spatial frequency restricted by the objective aperture and the damping of the wave function due to limited coherence. It may be denoted as the product of an aperture function A_{1} and a damping factor A(k), where the aperture function is given by:

`A_{1} = 1\space for \space k \lt k_{max} \space and \space A_{1}=0 \space for \space k \ge k_{max} `

k_{max} is the maximum spatial frequency, which can be transferred.

## Contrast transfer function versus spatial frequency k, without damping

Every zero-crossing of the graph corresponds to a contrast inversion in the image. Up to the first zero-crossing k_{0} (in our picture about 4.1/nm) the contrast does not change its sign. The reciprocal value 1/ k_{0} is called the Scherzer point resolution (it corresponds to the radius of the first dark ring in the diffractogram). The defocus value which - spherical aberration given - maximizes this point resolution is called the Scherzer focus. Working in the Scherzer focus ensures the transmission of a broad band of spatial frequencies with constant contrast and allows an unambiguous interpretation of the image.

The value of the Scherzer focus can be estimated as

`\Delta{f}_{scherzer}=1.2\cdot\sqrt{\lambda\cdot{c_s}}`

so that the point resolution is given by

`\delta=0.64\cdot\lambda^\frac{3}{4}\cdot{c_s^\frac{1}{4}}`

## Contrast transfer as a function of spatial frequency k, with chromatic aberration

Since every electron source has a finite energy width the electrons undergo different phase shifts when passing through the objective lens. The highest possible phase shift is given by the energy width of the electron source.

`\phi(k)=\pi\cdot{c_c}\cdot\frac{\Delta{E}}{E_\omicron}\cdot\lambda\cdot{k^2}`

The resulting damping of the phase contrast transfer function is

`A(k)=exp(-0.5\cdot\pi^2\cdot{c^2_c})\cdot(\frac{\Delta{E}}{E_\omicron})^2\cdot\lambda^2\cdot{k^4})`

This damping resulting from chromatic aberration limits the spatial frequency maximally transferred. It is the reciprocal value of the minimum distance in the object, which can be resolved. This distance is commonly known as the information limit. The contrast of the information limit is defined as damped by a factor of 1/e^{2} compared to the case without damping. In the picture 1/e^{2} is represented by the horizontal line. In the case depicted here the information limit is about (1/8.3) nm.

Optimizing the information limit is possible by correcting both spherical AND chromatic aberration, this allows the resolution of even smaller object details. The following pictures are calculated diffractograms. In the left one only the spherical aberration is corrected, in the right one both the spherical and the chromatic aberration are corrected. The Moiré patterns are artefacts due to the scanning mode of the monitor.

## Advantage of correction

![]()
In every uncorrected electron microscope the reachable point resolution is much worse than the optimum information limit. **But using an electron microscope with correction allows for optimizing the spherical aberration coefficient and the defocus so that the point resolution equals the information limit.**

The attainable information limit results from the chromatical aberration (graph of the contrast transfer function with damping). Using equation (4) delivers the coefficent of spherical aberration. For this example c_{s}=0.080 mm. The optimum defocus can be calculated using this value in equation (3) to be 17.01 nm.

Inserting these values into the contrast transfer function results in the following graph:

Correcting the spherical aberration allows for increasing the point resolution until it equals the information limit, in this case (1/8.3) nm = 0.12 nm.

## The low frequency limit: phaseplates as an alternative

As can be seen from the phase contrast transfer function, aberration correction provides a way to push the point resolution to higher spatial frequencies. However, in both cases - corrected or uncorrected - the spatial frequencies below 1/nm show no phase contrast. The ratio between the highest and lowest spatial frequency showing phase contrast is well below 10.

The following simulated images of a macroscopic object demonstrate the resulting imaging quality of the observed specimen. Let us assume that this image represents a microscopic phase object with a size of 16x16 nm. The colour white represents a phase shift of π/10 and black corresponds to a phase shift of 0. Thus, the image shows a weak phase object.

If such an object is imaged with a phase contrast transfer function (PCTF) of an uncorrected microscope, the result is very disappointing. The coarse structure of the specimen can hardly be recognised. Only the edges of the sample are visible, but the low frequency components are lost and lots of artefacts are produced. But even for the Cs corrected PCTF the image only shows the fine structure of the sample. The number of artefacts, however, is reduced. But any coarse structure is still lost due to the low frequency cut-off. For many material science applications this is acceptable, because long range order phenomena are less important, but for the life sciences a cut-off at about 1 nm is crucial.

Phase contrast imaging of an original weak phase object in an uncorrected and a C_{s}-corrected microscope and with a Boersch or a Zach phase plate.

To overcome this obstacle, phaseplates have been developed. A phase plate in electron optics shifts the phase of the unscattered electrons (the so-called zero beam) relative to the scattered ones. In the diffraction plane electrons with different scattering angles pass the plane at different positions. So applying a voltage of a few millivolts only to the zero beam in this plane achieves the desired phase shift of π/2.

A Boersch phase plate (picture from KIT) and a Zach phase plate (picture from Katrin Schultheiss's Diss (12MB full text in German, very interesting!).

One way to reach this goal is the so-called Boersch phase plate: A micron-sized structure applies the voltage in a small bore of the phase plate only to the unscattered beam, which passes through this central hole. The scattered electrons are unaffected. A variant of this type of phase plate is the so-called Zach plate as patented by CEOS, where the voltage is applied in the core of a coaxial structure, which approaches the unscattered beam from one side. This variant avoids some obstruction of electrons with small scattering angles by the ring-shaped structure of the Boersch plate.

Several groups around the world try to produce efficient phase plates and use them for imaging. For further reading please consult the following links:

- Phase plate development at the Karlsruher Institut für Technologie
- An article about phase plates for biological applications in bioPHOTONICS

Some original papers about phase plates can be found in our literature list.