Aberrations Astigmatism

Once the primary electron beam is created, it must be demagnified with condenser lenses and then focused onto the sample with objective lenses. These electron lenses are electromagnetic in nature and use electric and magnetic fields to steer the electrons. Such lenses are subject to severe spherical and chromatic aberrations. Therefore, a point primary beam source is blurred into a primary beam disk to an extent dependent on the energy and energy spread of the primary electrons. In addition, these lenses are also subject to astigmatism. AH three of these effects ultimately limit the primary beam spot size and hence, the lateral resolution achievable with sem. 

Lastly, it is possible to correct for all three major aberrations. Spherical, coma and astigmatism. This gives a system with very large FOV, but with other less desirable features. Examples are ... 

So far, only defocus and spherical aberration have been considered as aberrations affecting the image contrast. Both depend only on the magnitude of the spatial frequency g[ but not on the diffraction direction, thus resulting in rotationally symmetric phase shifts (Equation 11). However, the objective lens may exhibit further aberrations resulting in additional phase shifts, which are not necessarily rotational symmetric. The most important of these additional aberrations are astigmatism and coma. 

Several other approaches have been followed towards quantitative HRTEM imaging. One approach is the development of new hardware to correct for or alleviate some of the aberrations in the image, e.g. spherical aberration corrector (Rose 1990, Haider et al. 1995) and three-fold astigmatism corrector (Overwijk et al. 1997). An alternative approach is the development of new methods to retrieve the exit wave function, e.g. off-axis holography (Eichte 1986, Lichte and Rau 1994) and focal-series reconstruction (FSR) (Coene et al. 1992, 1996, Thust et al. 1996a). While each approach has its distinct advantages, we are only going to discuss focal-series reconstruction in this paper. 

The only practical difficulty comes from determining the appropriate aberration correction. In Truelmage, this has been automated, e.g. for the defocus and the 2-fold astigmatism. For a weak phase object, the real part of exit wave function should be constant (Equation 2). Thus, the defocus and 2-fold astigmatism can be determined by minimizing the contrast of the real part of the exit-wave function. 

Figure 10. Reconstructed exit wave function (phase) of the same area as shown in Figure 5. The left image shows the phase of the exit wave function before aberration correction and the right image shows the phase after correction for residual 2-, 3-fold astigmatism, and coma. The difference clearly illustrates that numerical correction of residual aberrations is crucial for...

Like their glass analogues, magnetic lenses suffer from such defects as coma, distortion, astigmatism, and chromatic and spherical aberration. 

Astigmatism is due to asymmetry of the magnetic field about the axis of the lens. Fortunately, this aberration can be corrected by means of Astig-mator, which superimposes an additional magnetic field (whose strength and direction can be controlled by the microscopist) across the gap in the pole piece of the lens. 

As with optical lenses, electromagnetic lenses have aberrations (chromatic aberration, spherical aberration, electron diffraction limit, astigmatism, etc.) each entailing an enlargement of the electron probe expressed by the diameter of a circle of least confusion. Under standard operating conditions, when the astigmatism has been adjusted, only spherical aberration plays a significant role. The expression of the probe diameter becomes ... 

The absolute size of the aberrations inereases with the size of the lens and its f number. Moreover, oblique beams experienee eoma and astigmatism, i.e. the spot where the rays are eombined is no longer eireular. Moreover, the surfaee of best focus is curved. 

In TCSPC systems, light is often to be eolleeted from a small spot in a sample and transferred to a small-area deteetor, the slit of a monoehromator, or an optical fibre. Most optical systems for TCSPC do not use strongly oblique beams so that eoma, astigmatism, and curvature of the image field are usually not problems. However, to obtain a high colleetion effieieney, high f numbers must be used, and problems arise from spherical and chromatical aberration. 

The object of an ideal optical configuration is that there are no optical aberrations, no stray light, and no energy loss through the system. In practice, various compromises have to be made to minimize the effect of the various variables. Within a practical optical system, five types of optical aberration are possible which give rise to image distortion, namely primary spherical aberration, coma, astigmatism (anamorphism), field curvature, and distortion. 

Littrow Simple arrangement 1 mirror 3 reflections Slit positions allow compact system and minimize astigmatism Susceptible to coma aberration Needs paraboloid mirror Slits are close together The cunrature of the slit image is wavelength dependent... 

Monk-Gilleson Coma compensation and astigmatism correction 2 reflections 1 mirror High energy throughput Requires large grating Subject to linear shifts in aberration... 

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