Weierstrass lens

Figure 3.5-9 Principle of the Weierstrass lens The refractive index of a sphere with a radius r is n. A beam which would have — without the sphere — a focus at a distance of n r from the center of the sphere is focused inside the sphere at a distance of r/n from the center.

Beam condensers, by using a pair of ellipsoid mirrors, produce very small images of the Jacquinot stop or the entrance slit at the sample position. The size of these images may be even further reduced by making use of a Weierstrass sphere. Weierstrass (1856) showed that a spherical lens has two aplanatic points . If a sphere of a glass with a refractive index n is introduced into an optical system which has a focus at a distance of r n from its center, then the beam is focused inside the sphere at a distance of r/n from the center (Fig. 3.5-9). In this case the angle O in Eq. 3.4-5 may approach 90°. Thus, a sample with a very small area can fully fit the optical conductance of the spectrometer (Fig. 3.4-2d). Microscopes usually have an optical conductance which is considerably lower than that of spectrometers. In this case, the sample and the objective are the elements limiting the optical conductance (Schrader, 1990 Sec. 3.5.3.3). 

Past strategies for increasing the storage capacity of optical disks were based on a reduction of A and an increase in NA, as can be seen from Table 12.1. In principle, a reduction in the spot size can be achieved with the aid of solid immersion lenses. This as yet not practically exploited technique, operating with a hemispherical or a Weierstrass superspherical lens placed near the recording medium (< 100 nm), yields a reduced spot size, S, as is evident from Eqs. (12-2) and (12-3), respectively, where n denotes the refractive index of the lens [8],... 

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