p. 354); the argument is based on the principle that the optical
distance from object to image is constant.
Fig. 14.
“Taking the case of a convex lens of glass, let us suppose that parallel rays DA, EC, GB (fig. 14) fall upon the lens ACB, and are collected by it to a focus at F. The points D, E, G, equally distant from ACB, lie upon a front of the wave before it impinges upon the lens. The focus is a point at which the different parts of the wave arrive at the same time, and that such a point can exist depends upon the fact that the propagation is slower in glass than in air. The ray ECF is retarded from having to pass through the thickness (d) of glass by the amount (n − 1)d. The ray DAF, which traverses only the extreme edge of the lens, is retarded merely on account of the crookedness of its path, and the amount of the retardation is measured by AF−CF. If F is a focus these retardations must be equal, or AF − CF = (n − 1)d. Now if y be the semi-aperture AC of the lens, and f be the focal length CF, AF − CF = √(f2 + y2) − f = 12y2/f approximately, whence
| f = 12y2 / (n − 1)d. | (12) |
In the case of plate-glass (n − 1) = 12 (nearly), and then the rule (12) may be thus stated: the semi-aperture is a mean proportional between the focal length and the thickness. The form (12) is in general the more significant, as well as the more practically useful, but we may, of course, express the thickness in terms of the curvatures and semi-aperture by means of d = 12y2 (r1−1 − r2−1). In the preceding statement it has been supposed for simplicity that the lens comes to a sharp edge. If this be not the case we must take as the thickness of the lens the difference of the thicknesses at the centre and at the circumference. In this form the statement is applicable to concave lenses, and we see that the focal length is positive when the lens is thickest at the centre, but negative when the lens is thickest at the edge.”
Regulation of the Rays.
The geometrical theory of optical instruments can be conveniently divided into four parts: (1) The relations of the positions and sizes of objects and their images (see above); (2) the different aberrations from an ideal image (see Aberration); (3) the intensity of radiation in the object- and image-spaces, in other words, the alteration of brightness caused by physical or geometrical influences; and (4) the regulation of the rays (Strahlenbegrenzung).
Fig. 15.
The regulation of rays will here be treated only in systems free from aberration. E. Abbe first gave a connected theory; and M. von Rohr has done a great deal towards the elaboration. The Gauss cardinal points make it simple to construct the image of a given object. No account is taken of the size of the system, or whether the rays used for the construction really assist in the reproduction of the image or not. The diverging cones of rays coming from the object-points can only take a certain small part in the production of the image in consequence of the apertures of the lenses, or of diaphragms. It often happens that the rays used for the construction of the image do not pass through the system; the image being formed by quite different rays. If we take a luminous point of the object lying on the axis of the system then an eye introduced at the image-point sees in the instrument several concentric rings, which are either the fittings of the lenses or their images, or the real diaphragms or their images. The innermost and smallest ring is completely lighted, and forms the origin of the cone of rays entering the image-space. Abbe called it the exit pupil. Similarly there is a corresponding smallest ring in the object-space which limits the entering cone of rays. This is called the entrance pupil. The real diaphragm acting as a limit at any part of the system is called the aperture-diaphragm. These diaphragms remain for all practical purposes the same for all points lying on the axis. It sometimes happens that one and the same diaphragm fulfils the functions of the entrance pupil and the aperture-diaphragm or the exit pupil and the aperture-diaphragm.
Fig. 15 shows the general but simplified case of the different diaphragms which are of importance for the regulation of the rays. S1, S2 are two centred systems. A′ is a real diaphragm lying between them. B1 and B′2 are the fittings of the systems. Then S1 produces the virtual image A of the diaphragm A′ and the image B2 of the fitting B′2, whilst the system S2 makes the virtual image A″ of the diaphragm A′ and the virtual image B′1 of the fitting B1. The object-point O is reproduced really through the whole system in the point O′. From the object-point O three diaphragms can be seen in the object-space, viz. the fitting B1, the image of the fitting B2 and the image A of the diaphragm A′ formed by the system S1. The cone of rays nearest to B2 is not received to its total extent by the fitting B1, and the cone which has entered through B1 is again diminished in its further course, when passing through the diaphragm A′, so that the cone of rays really used for producing the image is limited by A, the diaphragm which seen from O appears to be the smallest. A is therefore the entrance pupil. The real diaphragm A′ which limits the rays in the centre of the system is the aperture diaphragm. Similarly three diaphragms lying in the image-space are to be seen from the image-point O′—namely B′, A″, and B′2. A″ limits the rays in the image-space, and is therefore the exit pupil. As A is conjugate to the diaphragm A′ in the system S1, and A″ to the same diaphragm A′ in the system S2, the entrance pupil A is conjugate to the exit pupil A″ throughout the instrument. This relation between entrance and exit pupils is general.
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| Fig. 16. |
 |
| Fig. 17a. Fig. 17b. |
The apices of the cones of rays producing the image of points near the axis thus lie in the object-points, and their common base is the entrance pupil. The axis of such a cone, which connects the object point with the centre of the entrance pupil, is called the principal ray. Similarly, the principal rays in the image-space join the centre of the exit pupil with the image-points. The centres of the entrance and exit pupils are thus the intersections of the principal rays.
For points lying farther from the axis, the entrance pupil no longer alone limits the rays, the other diaphragms taking part. In fig. 16 only one diaphragm L is present besides the entrance pupil A, and the object-space is divided to a certain extent into four parts. The section M contains all points rendered by a system with a complete aperture; N contains all points rendered by a system with a gradually diminishing aperture; but this diminution does not attain the principal ray passing through the centre C. In the section O are those points rendered by a system with an aperture which gradually decreases to zero. No rays pass from the points of the section P through the system and no image can arise from them. The second diaphragm L therefore limits the three-dimensional object-space containing the points which can be rendered by the optical system. From C through this diaphragm L this three-dimensional object-space can be seen as through a window. L is called by M von Rohr the entrance luke. If several diaphragms can be seen from C, then the entrance luke is the diaphragm which seen from C appears the smallest. In the sections N and O the entrance luke also takes part in limiting the cones of rays. This restriction is known as the “vignetting” action of the entrance luke. The base of the cone of rays for the points of this section of the object-space is no longer a circle but a two-cornered curve which arises from the object-point by the projection of the entrance luke on the entrance pupil. Fig. 17a shows the base of such a cone of rays. It often happens that besides the entrance luke, another diaphragm acts in a vignetting manner, then the operating aperture of the cone of rays is a curve made up of circular arcs formed out of the entrance pupil and the two projections of the two acting diaphragms (fig. 17b).
If the entrance pupil is narrow, then the section NO, in which the vignetting is increasing, is diminished, and there is really only one division of the section M which can be reproduced, and of the section P which cannot be reproduced. The angle w + w = 2w, comprising the section which can be reproduced, is called the angle of the field of view on the object-side. The field of view 2w retains its importance
