Electron microscopy

Electron microscopes are microscopes that use electrons as the illuminating source instead of visible light, with the same basic principles as a light microscope. This is due to the fact that that electrons exhibit wave-particle duality. The discussion here is limited to transmis- sion electron microscopes (TEM) that are used to study thin semi-transparent specimens, particularly biological specimens. A diagram of a TEM is shown in Fig. 1.8. An advantage of using electrons is that their wavelength is small, of the order of 0.01 to 0.002nm, so that a very high resolution is potentially available. Another advantage of electron microscopy is that “electron lenses” can be formed by magnetic fields produced using electromagnets. However, in practice resolution is limited by aberrations due to imperfect magnetic lenses. The resolution obtainable varies, but is typically limited to about 0.1-0.5nm [43, 44, 45]. The basic principle of a transmission electron microscope is described as follows (Fig. 1.9). Electrons emitted from an electron gun first travel through a vacuum column in the mi- croscope (as electrons are easily scattered by air), and then pass through a condenser lens before encountering the specimen. The electron wave on passing through the specimen suffers an attenuation and phase shift, and the refractive index of the specimen is therefore complex. The specimen must be thin to reduce multiple scattering and excessive attenua- tion. The scattered electrons, or the “exit wave”, then pass though the objective lens which forms the first intermediate image. Subsequently, the image is enlarged by the following intermediate (and projector) lenses and finally the data is recorded/displayed as electrons hitting a fluorescent screen, photographic plate, or light sensitive sensor (e.g., CCD cam- era).

Information in an electron microscope can be measured in the image plane or the diffrac- tion pattern can be recorded, depending on the configuration of the lenses. This is one of

1.6 Diffraction imaging 27

the main advantages of TEM. The main principles of electron microscopy can be illustrated by geometrical optics, and are shown in Fig. 1.9. In both the image and diffraction mode, a diffraction pattern is firstly formed by the objective lens at its back focal plane. In image mode (Fig. 1.9), an objective aperture is placed at the back focal plane of the objective lens to select the electron beam for the final image and intermediate image is formed in the im- age plane. A bright field image is formed if the undiffracted electrons pass through, and a dark field image is formed if the undiffracted electrons are blocked. In diffraction mode (Fig. 1.9), the intermediate lens is adjusted such that it is focused on the back focal plane of the objective lens and the diffraction pattern is magnified by the following lenses. An intermediate aperture can be placed at the image plane of the objective lens to limit the diffraction pattern to a selected area of the specimen, and this is referred to as selected area electron diffraction (SAED). Furthermore, by tilting the specimen over various angles with respect to the incident electron beam, diffraction patterns of different orientations can be recorded, allowing reconstruction of a three-dimensional (thick) specimen.

In TEM, the diffracted field can be approximated by Fraunhofer diffraction, and the wave function in the image plane and in the diffraction pattern, denotedf(x, y)andF(u, v) re- spectively, form a Fourier transform pair. In electron microscopy, bothf(x, y)andF(u, v) are complex, both containing the same information, and the phase information of each is lost, i.e., only|f(x, y)_|and|F(u, v)_|can be recorded. In practice, the resolution in image space is further degraded by aberrations, mainly caused by the lenses (other aberrations include changes in the energy level of the electrons and changes in the strength of magnetic fields) [45, 47]. These aberrations affect only the phase in Fourier space, i.e., an unknown phase term is introduced in the CTF, so that it is possible to recover a higher resolution image using the amplitude of the diffraction pattern if the phase information can be recov- ered.

Recovering the complex imagef(x, y)from measurement of its amplitude|f(x, y)_|and the amplitude of its Fourier transform|F(u, v)_|is a variation on the phase problem described in Section 1.6.1, and was first studied by Gerchberg and Saxton [48, 49]. Iterative transform algorithms similar to those described in Section 1.6.1 can be applied in which the known amplitudes in each domain are applied. Furthermore, the resolution of the reconstructed image can be extended beyond that recorded in the image plane by also estimating the amplitudes in the image using the high resolution diffraction amplitudes. If the specimen is thin enough that it affects only the phase of the diffracted field (known as a phase object), then the image amplitude is known to be constant, simplifying the problem.

An alternative approach is to recover the phase by analysing a set of defocused images directly using the transport of intensity equation (TIE). The TIE is a wave propagation equation derived by Teague [50] in terms of the intensity and phase under the small angle

1.6 Diffraction imaging 29

approximation and shows that the phase can be estimated from the intensity measure- ments in the Fresnel region. This method has been applied successfully [51, 52].