High-resolution transmission electron microscopy

The phase contrast transfer function is a function of limiting apertures and aberrations in the imaging lenses of a microscope. It describes their effect on the phase of the exit wave φ_e(x,u) and propagates it to the image wave. Following Williams and Carter,[6] if we assume the weak phase object approximation holds (thin sample) the contrast transfer function becomes

${\displaystyle CTF(u)=A(u)E(u)2\sin(\chi (u))}$

where A(u) is the aperture function, E(u) describes the attenuation of the wave for higher spatial frequency u, also called envelope function. χ(u) is a function of the aberrations of the electron optical system.

The last, sinusoidal term of the contrast transfer function will determine the sign with which components of frequency u will enter contrast in the final image. If one takes into account only spherical aberration to third order and defocus, χ is rotationally symmetric about the optical axis of the microscope and thus only depends on the modulus u = |u|, given by

${\displaystyle \chi (u)={\frac {\pi }{2}}C_{s}\lambda ^{3}u^{4}-\pi \Delta f\lambda u^{2}}$

where C_s is the spherical aberration coefficient, λ is the electron wavelength, and Δf is the defocus. In transmission electron microscopy, defocus can easily be controlled and measured to high precision. Thus one can easily alter the shape of the contrast transfer function by defocusing the sample. Contrary to optical applications, defocusing can increase the precision and interpretability of the micrographs.

The aperture function cuts off beams scattered above a certain critical angle (given by the objective pole piece for ex), thus effectively limiting the attainable resolution. However it is the envelope function E(u) which usually dampens the signal of beams scattered at high angles, and imposes a maximum to the transmitted spatial frequency. This maximum determines the highest resolution attainable with a microscope and is known as the information limit. E(u) can be described as a product of single envelopes:

${\displaystyle E(u)=E_{s}(u)E_{c}(u)E_{d}(u)E_{v}(u)E_{D}(u),\,}$

due to

E_s(u): angular spread of the source

E_c(u): chromatic aberration

E_d(u): specimen drift

E_v(u): specimen vibration

E_D(u): detector

Specimen drift and vibration can be minimized in a stable environment. It is usually the spherical aberration C_s that limits spatial coherency and defines E_s(u) and the chromatic aberration C_c, together with current and voltage instabilities that define the temporal coherency in E_c(u). These two envelopes determine the information limit by damping the signal transfer in Fourier space with increasing spatial frequency u

${\displaystyle E_{s}(u)=\exp \left[-\left({\frac {\pi \alpha }{\lambda }}\right)^{2}\left({\frac {\delta \mathrm {X} (u)}{\delta u}}\right)^{2}\right]=\exp \left[-\left({\frac {\pi \alpha }{\lambda }}\right)^{2}(C_{s}\lambda ^{3}u^{3}+\Delta f\lambda u)^{2}\right],}$

where α is the semiangle of the pencil of rays illuminating the sample. Clearly, if the wave aberration ('here represented by C_s and Δf) vanished, this envelope function would be a constant one. In case of an uncorrected transmission electron microscope with fixed C_s, the damping due to this envelope function can be minimized by optimizing the defocus at which the image is recorded (Lichte defocus).

The temporal envelope function can be expressed as

${\displaystyle E_{c}(u)=\exp \left[-{\frac {1}{2}}\left(\pi \lambda \delta \right)^{2}u^{4}\right],}$.

Here, δ is the focal spread with the chromatic aberration C_c as the parameter:

${\displaystyle \delta =C_{c}{\sqrt {4\left({\frac {\Delta I_{\text{obj}}}{I_{\text{obj}}}}\right)^{2}+\left({\frac {\Delta E}{V_{\text{acc}}}}\right)^{2}+\left({\frac {\Delta V_{\text{acc}}}{V_{\text{acc}}}}\right)^{2}}},}$

The terms ${\displaystyle \Delta I_{\text{obj}}/I_{\text{obj}}}$ and ${\displaystyle \Delta V_{\text{acc}}/V_{\text{acc}}}$ represent instabilities in of the total current in the magnetic lenses and the acceleration voltage. ${\displaystyle \Delta E/V_{\text{acc}}}$ is the energy spread of electrons emitted by the source.

The information limit of current state-of-the-art transmission electron microscopes is well below 1 Å. The TEAM project at Lawrence Berkeley National Laboratory resulted in the first transmission electron microscope to reach an information limit of <0.5 Å in 2009 [7] by the use of a highly stable mechanical and electrical environment, an ultra-bright, monochromated electron source and double-hexapole aberration correctors.

contrast transfer function of the OAM microscope

Choosing the optimum defocus is crucial to fully exploit the capabilities of an electron microscope in high resolution transmission electron microscopy mode. However, there is no simple answer as to which one is the best.

In Gaussian focus one sets the defocus to zero, the sample is in focus. As a consequence contrast in the image plane gets its image components from the minimal area of the sample, the contrast is localized (no blurring and information overlap from other parts of the sample). The contrast transfer function becomes a function that oscillates quickly with C_su⁴. This means that for certain diffracted beams with a spatial frequency u the contribution to contrast in the recorded image will be reversed, thus making interpretation of the image difficult.

In Scherzer defocus, one aims to counter the term in u⁴ with the parabolic term Δfu² of χ(u). Thus by choosing the right defocus value Δf one flattens χ(u) and creates a wide band where low spatial frequencies u are transferred into image intensity with a similar phase. In 1949, Scherzer found that the optimum defocus depends on microscope properties like the spherical aberration C_s and the accelerating voltage (through λ) in the following way:

${\displaystyle \Delta f_{\text{Scherzer}}=-1.2{\sqrt {C_{s}\lambda }}\,}$

where the factor 1.2 defines the extended Scherzer defocus. For the CM300 at NCEM, C_s = 0.6mm and an accelerating voltage of 300keV (λ = 1.97 pm) (Wavelength calculation) result in Δf_Scherzer = -41.25 nm.

The point resolution of a microscope is defined as the spatial frequency u_res where the contrast transfer function crosses the abscissa for the first time. At Scherzer defocus this value is maximized:

${\displaystyle u_{\text{res}}({\text{Scherzer}})=0.6\lambda ^{3/4}C_{s}^{1/4},}$

which corresponds to 6.1 nm⁻¹ on the CM300. Contributions with a spatial frequency higher than the point resolution can be filtered out with an appropriate aperture leading to easily interpretable images at the cost of a lot of information lost.

Gabor defocus is used in electron holography where both amplitude and phase of the image wave are recorded. One thus wants to minimize crosstalk between the two. The Gabor defocus can be expressed as a function of the Scherzer defocus as

${\displaystyle \Delta f_{\text{Gabor}}=0.56\Delta f_{\text{Scherzer}}}$

To exploit all beams transmitted through the microscope up to the information limit, one relies on a complex method called exit wave reconstruction which consists in mathematically reversing the effect of the contrast transfer function to recover the original exit wave φ_e(x,u). To maximize the information throughput, Hannes Lichte proposed in 1991 a defocus of a fundamentally different nature than the Scherzer defocus: because the dampening of the envelope function scales with the first derivative of χ(u), Lichte proposed a focus minimizing the modulus of dχ(u)/du[8]

${\displaystyle \Delta f_{\text{Lichte}}=-0.75C_{s}(u_{\max }\lambda )^{2},}$

where u_max is the maximum transmitted spatial frequency. For the CM300 with an information limit of 0.8 Å Lichte defocus lies at −272 nm.

Exit wave reconstruction through focal series

To calculate back to φ_e(x,u) the wave in the image plane is back propagated numerically to the sample. If all properties of the microscope are well known, it is possible to recover the real exit wave with very high accuracy.

First however, both phase and amplitude of the electron wave in the image plane must be measured. As our instruments only record amplitudes, an alternative method to recover the phase has to be used. There are two methods in use today:

• Holography, which was developed by Gabor expressly for transmission electron microscopy applications, uses a prism to split the beam into a reference beam and a second one passing through the sample. Phase changes between the two are then translated in small shifts of the interference pattern, which allows recovering both phase and amplitude of the interfering wave.
• Through focal series method takes advantage of the fact that the contrast transfer function is focus dependent. A series of about 20 pictures is shot under the same imaging conditions with the exception of the focus which is incremented between each take. Together with exact knowledge of the contrast transfer function, the series allows for computation of φ_e(x,u) (see figure).

Both methods extend the point resolution of the microscope past the information limit, which is the highest possible resolution achievable on a given machine. The ideal defocus value for this type of imaging is known as Lichte defocus and is usually several hundred nanometers negative.