Experimental Resolution of Material Features

The factor 1.22 in Eq. 2.1 was empirically derived by Rayleigh. It may be derived from the radius of the circle, known as the Airy disk, from the optical transfer function. In 1873, the German physicist Ernst Karl Abbe (1840-1905) showed that the numerical 

In this equation, n is the refractive index of the imaging medium and 6 is the angular aperture, the apparent angle of the lens apeiTure as seen from the focal point. The human eye has an entrance pupil of 7 mm and an NA value of 0.002. Thus, by Eq. 2.1, the human eye can resolve two objects, at a normal viewing distance and when the objects are 

In addition to lateral resolution, different depth resolutions are obtained in the various techniques, which depend on the penetration depth of the probing beam. The penetration depth is a function of the beam energy, but is also material dependent. Some sources, together with their wavelengths and their typical depth and lateral resolutions, are hsted in Table 2.1. 

To be examined by ophcal microscopy, a material must, of course, be opaque to visible hght, for its surface to be observed. Contrasts in the produced image are as a result of differences in the reflechvity of the various regions of the microstmcture. Given the penetrahon depths hsted in Table 2.1, it is obvious that not only light, but also electrons, probe just the surfaces (the top-most atomic layers), whereas neutrons and X-rays provide information about the bulk. Hence, light and electron beams are used in microscopes for examining solid surfaces. Because it is well known that the surface crystalline stmc-ture of a sohd may differ from that of the bulk, the surfaces of most samples are usually 

TABLE 2.1. Some Types of Probes Used in Materials Characterization 

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