Image formation with lenses

Figure 2.1. Schematic of image formation with a lens. Radiation from any single point on the object is brought to a focus at a point in the image plane (ip), forming an image. At the back focal plane (bfp), radiation leaving anywhere on the object in a single direction is brought to a single point.

Figures 5e and 5f show OCT images of two flax seeds (GMF) 5 minutes after being soaked in water. Multiple grouping of water transport channels with formation of lens-shaped water-bearing structures can be clearly distinguished.

Real image formation for a biconvex lens (positive, converging) with refractive index n>1. As long as u>f, a real image forms, upside down, on the opposite side of the lens from the object. 

Chapter 1 is concerned with the fundamental principles of image formation by a lens. These principles were first formulated by Ernst Abbe in 1873 and are basic to the chapters that follow. According to the Abbe theory, the image of an illuminated object is the result of a twofold diffraction process. First, the Fraunhofer diffraction pattern of the object is formed in the back focal plane of the lens. Second, the light waves travel... 

Now we must examine the physics of image formation by a lens more closely, and to do this we must introduce the ideas of Fourier optics. It would be inappropriate here to develop fully these ideas, as is done in most modern textbooks on optics, but it is important to understand clearly the fundamental concepts of Fourier optics because we shall need them when we deal specifically with electron diffraction. However, before continuing, it is necessary to digress briefly to introduce the mathematics used to describe plane and spherical waves. 

This chapter describes briefly the basic construction and characteristics of the modern transmission electron microscope and discusses its principal modes of operation. Because the electron microscope is an analogue of the optical (or light) microscope, we also consider briefly the basic features of the optical microscope this will also provide a link with our earlier discussion of the optical principles of image formation by a lens. 

Figure 14.1. Schematic diagram showing the principle of image formation and diffraction in the transmission electron microscope. The incident beam/o illuminates the specimen. Scattered and unscattered electrons are collected by the objective lens and foeused back to form first an electron diffraction pattern and then an image. For a 2D or 3D crystal, the electron-diffraetion pattern would show a lattice of spots, eaeh of whose intensity is a small fraetion of that of the incident beam. In praetiee, an in-focus image has no eontrast, so images are recorded with the objeetive lens slightly defocused to take advantage of the out-of-focus phase-contrast mechanism.

We will now consider how to simulate this method of image formation in the X-ray diffraction experiment where we have to use a mathematical replacement for the objective lens. The studies by Porter are of great importance because they show how the Bragg reflections give the amplitude components of a Fourier series representing the electron density in the crystal (the electron-density map). In effect, Fourier analysis takes place in the diffraction experiment, so that the scattering of X rays by the electron density in the crystal produces Bragg reflections, each with a different amplitude F hkl) and relative phase Qhkl-... 

A Fourier synthesis is a mathematical calculation whereby, in the case ofX-ray diffraction, the scattered waves (with correct amplitudes and relative phases) are recombined to give the electron density in the crystal. It is essentially the opposite of a Fourier analysis and is the equivalent of image formation by a lens. It is the stage of the experiment in which the crystallographer and the computer act as the lens of a microscope. Provided the relative phases can be found, an electron density map can be calculated (Fig. 14). 

With a detector positioned in the lens (this type of detector is fitted on high performance field effect microscopes), only SE 1 -type secondary elections are involved in the formation of the image. The image then becomes characteristic of the surface regardless of the voltage (Fig. 7.7). It should, however, be noted that the contrast is better at 1 kV because the interaction volume is essentially contained within the carbon film. 

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