Image resolution

The resolution of an image is the smallest distance between two points at which they may be distinguished as separate. The resolution of perfect optical lenses is limited by diffraction effects: the finite size of the lens(aperture) causes a modulation of transmitted light intensity collected on a viewing screen some distance away. The pattern of intensity, known as an Airy pattern, displays a strong central maximum (i.e. the Airy disk), surrounded by concentric minima and maxima.

A similar effect can be expected for electron lenses in the TEM: the intensity transmitted by the objective lens will be affected by diffraction such that a point-like object in the specimen plane will produce an Airy disk in the image plane. Two point-like objects in the specimen will be distinguished as separate, if their distance \(r_d \leq {0.61 \lambda \over \alpha}\), where \(\lambda\) is the wavelength of the electron beam and \(\alpha\) is the semi-angle subtended by the lens(aperture). This can be defined as the resolution of a perfect electron lens, based on the Rayleigh criterion.

Electron lenses are not perfect. They suffer from astigmatism, as well as chromatic and spherical aberrations, which arise from the spread of electron velocities in the beam, their angular distribution, and their distance for the optic axis as they travel through the magnetic field generated by the lenses.

Lens astigmatism is corrected by adjusting lens stigmators to compensate image distortions.

The effect of chromatic aberrations is seen when electrons travelling at different velocities experience a different Lorentz force as they cross the lens, and are focused at different distances along the optic axis. This degrades the resolution of the image. The effect can be reduced substantially by using a FEG electron source with a small energy spread. It is important to note that the beam energy distribution always broadens when electrons interact with the specimen through inelastic collisions. Hence small chromatic distortions are unavoidable in TEM images.

A lens is said to display spherical aberration when the field of the lens behaves differently for electrons travelling near the optic axis, and those travelling off-axis. The image resolution is degraded by \(r_s = C_s \alpha^3 \) , where \(C_s\) is the spherical aberration coefficient (usually expressed in mm), and \(\alpha\) is, again, the semi-angle subtended by the lens(aperture). Spherical aberration may be reduced by forming images just with electrons that travel close to the optic axis, i.e. minimising \(\alpha\). As you can see in the animations this can be accomplished using a small aperture to exclude electron trajectories that cross the lens far from its centre.

Reducing the aperture size reduces the beam current and increase the diffraction experienced by the beam. There is, therefore, an optimum aperture size for the greatest resolution. The optimum resolution can be expressed as: \(r_{opt} = \lambda^{1/4}C_s^{3/4}\).

Conventional TEMs can achieve resolutions of 0.2 nm, and hence allow imaging of atomic lattices. Aberration corrected TEMs, where additional electron-optic components are introduced to compensate for spherical and chromatic aberrations, can achieve point resolutions below 0.1 nm (in phase contrast images).