Lens curvature

Curvature and rotation lenses correct for any imperfections (aberrations) in the cross-sectional shape of the beam before it reaches the collector slit. The curvature lens provides a means of changing any banana-shaped beam cross-section into a rectangular shape (Figure 24.8). The rotation lens rotates the beam such that the sides of the beam become parallel with the long axis of the collector slit (Figure 24.8). 

In real curvature sensors, a vibrating membrane mirror is placed at the telescope focus, followed by a collimating lens, and a lens array. At the extremes of the membrane throw, the lens array is conjugate to the required planes. The defocus distance can be chosen by adjusting the vibration amplitude. The advantage of the collimated beam is that the beam size does not depend on the defocus distance. Optical fibers are attached to the individual lenses of the lens array, and each fiber leads to an avalanche photodiode (APD). These detectors are employed because they have zero readout noise. This wavefront sensor is practically insensitive to errors in the wavefront amplitude (by virtue of normahzing the intensity difference). 

Activation of ocular parasympathetic fibers (see below) results in narrowing of the pupil and increased curvature of the lens, enabling near objects to be brought into focus (accommodation). 

The operation of a transmission scanning acoustic microscope requires the lenses to be set up so that they are accurately confocal. This requires holders that can be moved relative to each other along three axes, with rather fine adjustment, and that are rigid to better than a wavelength even when a specimen is vibrating between them. The separation must first be set. If ro is the radius of curvature of each lens, ciq the aperture radius, and n the refractive index, then the focal planes of the two lenses will coincide when the separation between their front surfaces is... 

Hence the radius of curvature of the lens surface can be found from the relationship... 

The effect of varying the distance between the lens and the transducer is to vary the illumination of the lens, and this in turn affects the point-spread function of the lens (Chou et al. 1988). Figure 4.3 shows the point-spread function calculated for three values of the transducer-lens spacing for the same lens with a radius of curvature r0 = 47Ao, where Ao is the wavelength... 

The form of this lens Is meniscus, or concavo-convex, the concave side being turned towards the objects and the radius of curvature of the image is equal to. the focal length of the lens, plus the radius of its convex surface it is, therefore, longer by the lattor quantity than the radius of curvature of the image formed by a small single lens without a diaphragm in front. 

In the preceding Figs, c denotes crown, f flint-glass. Both forms have equal radii of the front and bach surfaces, equal focal lengths, equal curvature of the image, and an equal distortion hut the plan shown in Fig. 409 is the best, because there is rather less spherical aberration. Even this is very bad, and should he superseded by the view-lens, which has recently been invented by Professor Petzyal, and which will be described presently. 

Real image" means that the image is on the same side of the mirror as the object). This relationship, known as the "thin-lens fomula," is approximate it holds exactly only when u = v = R, that is, when a point source and its image are both at the center of curvature C. Also, if the object is moved to I, then the image will form at O that is, the points O and I are "conjugate," and... 

Figure 3.1 Focusing properties (lens eiFect) of a homogeneous magnetic field for mass separation of ion beams. A, A" and C are the positions of the object, image and centre of curvature, respectively, and 1, " are the distances of object and image, respectively, from the boundaries of the magnetic field.

This result is the Kelvin equation. In this approximation, we have further assumed that the particle is essentially a flat particle (with a vapor pressure of Pq) compared to the radius of curvature of the neck. We can calculate the rate at which the neck increases by equating the rate of material transfer to the surface of the lens between the spheres with the increase in its volume. The rate of condensation, m, is proportional to the difference in equilibrium vapor pressure, ZiP, as given by... 

This pressure has the same value at all points deeper than c, and the term 2a IR is that due to the curvature of the surface, as found in 11. VIII G. If the surface of the liquid sphere X had been concave (e.g. an air bubble in water), the attraction of the lens-shaped part must obviously be subtracted instead of added. 

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