Lenses aberrations

Shifting the origin in the Fourier space by uci, we obtain the wave-function FT[0(r)]e > , from which the lens aberration term can be eliminated in principle by multiplication with the inverse of the aberration phase factor e . The inverse Fourier transform gives finally the amplitude and phase of the true object wave 0 (f). 

The micrograph or the image obtained on an EM screen, photographic film, or (more commonly today) a CCD is the result of two processes the interaction of the incident electron wave function with the crystal potential and the interaction of this resulting wave function with the EM parameters which incorporate lens aberrations. In the wave theory of electrons, during the propagation of electrons through the sample, the incident wave function is modulated by its interaction with the sample, and the structural information is transferred to the wave function, which is then further modified by the transfer function of the EM. 

H(u) is the Fourier Transform of h(r) and is called the contrast transfer function (CTF). u is a reciprocal-lattice vector that can be expressed by image Fourier coefficients. The CTF is the product of an aperture function A(u), a wave attenuation function E(u) and a lens aberration function B(u) = exp(ix(u)). Typically, a mathematical description of the lens aberration function to lowest orders builds on the Weak Phase Approximation and yields the expression ... 

Since Fp is a real quantity and thus iatV is imaginary, no contrast would be visible without the transfer function of the microscope (Equation 6). The lens aberrations result in a transfer of some imaginary part information at the exit plane into the real part at the image plane, which can be imaged by HRTEM. 

Equation 10 can be interpreted as the aberrations of the objective lens multiplying the intensities of the diffracted beams by a phase factor sin[2(g)], which depends on the spatial frequency. Thus, in the WPOA, the observed image is proportional to the projected potential, but is modulated by the phase factor. Without the phase shift, j, due to the lens aberrations, a weak phase object would not be visible in HRTEM (this is analogous to the interpretation of equation 6). 

Figure 1. Ray optical description of lens aberrations, (a) Perfect lens imaging a point P in the object plane onto a sharp point in the image plane, (b) Lens affected by spherical aberration, (c) Lens underfocused by a distance Z. (d) Lens overfocused by a distance Z. Spherical aberration and defocusing cause a blurring of the point P in the image plane.

Alternatively, one can measure these ionization cross-sections and use them in much the same way as k-factors [43] are used in EDX microanalysis. Two systematic measurements, one for K, L edges [44] and the other for M4,5 edges [45] are reported in the literature. However, experimentally measured cross-sections show large variations [34]. To overcome these difficulties, it is necessary to prepare standards of well known stoichiometry and to measure their thickness accurately. Finally, some of the measurements reported in the literature [33] have been made with large collection angles (-100 mrad) and should be used with caution as they could be erroneous due to lens aberration effects. 

The simplest microscope is the light microscope it consists of an objective lens and an eyepiece. Microscope objectives and eyepieces usually consist of complex lens systems of two or more lenses to correct for lens aberrations (Figure 4.6). The objective lens forms a real intermediate image, which is then magnified by the eyepiece. The objective lens and eyepiece are maintained at a fixed... 

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