DAF, which tra
verses only the ex
treme edge of the
lens, is retarded
B merely on account
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 semiaperture AC of the lens, and f be the focal length CF, AF—CF=~ (f2+y2)—f=aye/f approximately, whence
f=aye/(n—I)a. (12) In the case of plateglass (n — I) = (nearly), and then the rule (12) may be thus stated: the semiaperture 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 semiaperture by means of d = y2(ri—'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 afid at the circumference. In this farm 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: (I) 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 imagespaces, in other words, the alteration of brightness caused by physical or geometrical influences; and (4) the regulation of the rays (Strahlenbegrenzung).
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 objectpoints 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 imagepoint 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 innermostfulfils the functions of the entrance pupil and the aperturediaphragm or the exit pupil and the aperturediaphragm.
Fig. 15 shows the general but simplified case of the different diaphragms which are of importance for the regulation of the rays. SI, S2 are two centred systems. A' 'is a real diaphragm lying between them. Bl and B'2 are the fittings of the systems. Then S, 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 objectpoint 0 is reproduced really through the whole system in the point 0'. From the objectpoint 0 three diaphragms can be seen in the objectspace, viz. the fitting B1, the image of the fitting B2 and the image A of the diaphragm A' formed by the system Si. 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 0 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 imagespace are to be seen from the imagepoint O'—namely B', A", and B'2. A" limits the rays in the imagespace, and is therefore the exit pupil. As A is conjugate to the diaphragm A' in the system SI, 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.
The apices of the cones of rays producing the image of points near the axis thus lie in the objectpoints, 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 imagespace join the centre of the exit pupil with the imagepoints. 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 objectspace 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 0 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. FIG. 16.
The second diaphragm L therefore limits the threedimensional objectspace containing the points which can be rendered by the optical system. From C through this diaphragm L this threedimensional objectspace 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 0 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 objectspace is no longer a' circle but a twocornered curve which arises from the objectpoint by the projection of the entrance luke on the entrance pupil. Fig. r7a shows the base of such
a cone of rays. It often hap
pens that besides the entrance
luke, another diaphragm acts FIG, 17a. FIG. 17b. 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. rib).
If the entrance pupil is narrow, then the section NO, in which the vignetting is increasing, is diminished, and there isreally 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 objectside. The field of view 2w retains its importance
A'
and smallest ring is completely lighted, and forms the origin of the cone of rays entering the imagespace. Abbe called it the exit pupil. Similarly there is a corresponding smallest ring in the objectspace 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 aperturediaphragm. 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
if the entrance pupil is increased. It then comprises all points reached by principal rays. The same relations apply to the imagespace, in which there is an exit luke, which, seen from the middle of the exit pupil, appears under the smallest angle. It is the image of the entrance luke produced by the whole system. The imageside field of view 2w' is the angle comprised by the principal rays reaching the edge of the exit luke.
Most optical Instruments are used to observe objectreliefs (threedimensional objects), and generally an imagerelief (a threedimensional image) is conjugate to this objectrelief. It is sometimes required, however, to represent by means of an optical instrument the objectrelief on a plane or on a groundglass as in the photographic camera. For simplicity we shall assume the intercepting plane as perpendicular to the axis and shall call it, after von Rohr, the " ground glass plane." All points of the image not lying in this plane produce circular spots (corresponding to the form of the pupils) on it, which are called " circles of confusion." The groundglass plane (fig. 18) is conjugate to the objectplane E in the objectspace, perpendicular to the axis, and called the " plane focused for." All points lying in this plane are reproduced exactly on the groundglass plane as the points 00. The circle of confusion
Z on the plane focused for corresponds to the circle of confusion Z' on the groundglass plane. The figure formed on the plane focused for by the cones of rays from all of the objectpoints of the total objectspace directed to the entrance pupil, was called " objectside representation " (imago) by M von Rohr. This representation is a central projection. If, for instance, the entrance pupil is imagined so small that only the principal rays pass through, then they project directly, and the intersections of the principal rays represent the projections of the points of the object lying off the plane focused for. The centre of the projection or the perspective centre is the middle point of the entrance pupil C. If the entrance pupil is opened, in place of points, circles of confusion appear, whose size depends upon the size of the entrance pupil and the position of the objectpoints and the plane focused for. The intersection of the principal ray is the centre of the circle of confusion. The clearness of the representation on the plane focused for is of course diminished by the circles of confusion. This central projection does not at all depend upon the instrument, but is entirely geometrical, arising when the position and the size of the entrance pupil, and the position of the plane focused for have been fixed. The instrument then produces an image on the groundglass plane of this perspective representation on the plane focused for, and on account of the exact likeness which this image has to the objectside representation it is called the " representation copy." By moving it round an angle of 18o°, this representation can be brought into a perspective position to the objects, so that all rays coming from the middle of the entrance pupil and aiming at the objectpoints, would always meet the corresponding imagepoints. This representation is accessible to the observer in different ways in different instruments. If the observer desires a perfectly correct perspective impression of the objectrelief the distance of the pivot of the eye from the representation copy must be equal to the nth part of the distance of the plane focused for from the entrance pupil, if the instrument has produced a nth diminution of the objectside representation. The pivot of the eye must coincide with the centre of the perspective, because all images are observed in direct vision. It is known that the pivot of the eye is the point of intersection of all the directions in which one can look. Thus all these points represented by circles of confusion which are less than the angular sharpness of vision appear clear to the eye; the space containing all these objectpoints, which appear clear to the eye, is called the depth. The depth of definition, therefore, is not a special property of the instrument, but depends on the size of the entrance pupil, the position of the plane focused for and on the conditions under which the representation can be observed.
If the distance of the representation from the pivot of the eye be altered from the correct distance already mentioned, the angles of vision under which various objects appear are changed; perspective errors arise, causing an incorrect idea to be given of the depth. A simple case is shown in fig. 19. A cube is the object, and if it is observed as in fig. 19a with the representation copy at the correct distance, a correct idea of a cube will be obtained. If, as in figs. 19b and 19c, the distance is too great, there can be427
two results. If it is known that the farthest section is just as high as the nearer one then the cube appears exceptionally deepened, like a long parallelepipedon. But if it is known to be as deep as it is high then the eye will see it low at the back and high at the front. The reverse occurs when the distance of observation is too short, the body then appears either too flat, or the nearer sections seem too low in relation to those farther off. These perspective errors can be seen in any telescope. In the
('a) 6) cJ
After von Rohr.
telescope ocular the representation copy has to be observed under too large an angle or at too short a distance: all objects therefore appear flattened, or the more distant objects appear too large in comparison with those nearer at hand.
From the above the importance of experience will be inferred. But it is not only necessary that the objects themselves be known to the observer but also that they are presented to his eye in the customary manner. This depends upon the, way in which the principal rays pass through the system—in other words, upon the special kind of " transmission " of the principal rays. In ordinary vision the pivot of the eye is the centre of the perspective representation which arises on the very distant plane standing perpendicular to the mean direction of sight. In this kind of central projection all objects lying in front of the plane focused for are diminished when projected on this plane, and those lying behind it are magnified. (The distances are always given in the direction of light.) Thus the objects near to the eye appear large and those farther from it appear small. This perspective has been called by M von Rohr' "entocentric transmission " (fig. 20). If the entrance pupil of the instrument lies at infinity, then all the principal rays are parallel and the
After von Rohr. After von Rohr.
projections of all objects on the plane focused for are exactly as large as the objects themselves. After E. Abbe, this course of rays is called " telecentric transmission " (fig. 21). The exit pupil then lies in the imageside focus of the
system. If the perspective centre lies in front of the plane focused for, then the objects lying in front of this plane are
magnified and those behind it After von Rohr.
are diminished. This is just the FIG. 22.
reverse of perspective repre
sentation in ordinary sight, so that the relations of size and the arrangements for space must be quite incorrectly indicated (fig. 22) ; this representation is called by M von Rohr a " hypercentric transmission." (0. HR.)
End of Article: DAF 

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