Bible Encyclopedias
Lens

Forms of Lenses

By varying the signs and relative magnitude of the radii, lenses may be divided into two groups according to their action, and into four groups according to their form. According to their action, lenses are either co ll ecting, convergent C f' =-n'r/(n' - n) (8) f = nr/(n' -n). (9) By joining this simple refracting system with a similar one, so that the second spherical surface limits the medium of refractive index n', we derive the spherical lens. Generally the two spherical surfaces enclose a glass lens, and are bounded on the outside by air of refractive index 1.

The deduction of the cardinal points of a spherical glass lens in air from the relations already proved is readily effected if we regard the lens as a combination of two systems each having one refracting surface, the light passing in the first system from air to glass, and in the second from glass to air. If we know the refractive index of the glass n, the radii r 1, r 2 of the spherical surfaces, and the distances of the two lens-vertices (or the thickness of the lens d) we can determine all the properties of the lens. A biconvex lens is shown in fig. 8. Let F 1 be the first principal focus of the first system of radius r 1, and F 1 ' the second principal focus; and let S i be its vertex. Denote the distance F 1 S 1 (the first principal focal length) by f 1, and the corresponding distance F' 1 S 1 by f'1. Let the corresponding quantities in the second system be denoted by the same letters with the suffix By equations (8) and (9) we have I _ r 1 ,nr1 _ nr 2, r2 -I' f l_ -I' f - n-I' .f 2 n - I' r1 r2n (io) and condensing, or divergent and dispersing; the term positive is sometimes applied to the former, and the term negative to the latter. Convergent lenses transform a parallel pencil into a converging one, and increase the convergence, and diminish the divergence of any pencil. Divergent lenses, on the other hand, transform a parallel pencil into a diverging one, and diminish the convergence, and increase the divergence of any pencil. In convergent lenses the first principal focal distance is positive and the second principal focal distance negative; in divergent lenses the converse holds.

The four forms of lenses are interpretable by means of equation (io).

(n-I) {n (r2-ri) -f-d (n-I) } (I) If r I be positive and r 2 negative. This type is called biconvex (fig. 9, I). The first principal focus is in front of the lens, and the second principal focus behind the lens, and the two principal points FIG. 9.

are inside the lens. The order of the cardinal points is therefore FS 1 HH'S 2 F'. The lens is convergent so long as the thickness is less than n(ri-r2)/(n-I). The special case when one of the radii is infinite, in other words, when one of the bounding surfaces is plane is shown in fig. 9, 2. Such a collective lens is termed piano-convex. As d increases, F and H move to the right and F' and H' to the left. If d =n(r1-r2)/(n-I), the focal 'ength is infinite, i.e. the lens is telescopic. If the thickness be greater than n(r1-r2)/(n-I), the lens is dispersive, and the order of the cardinal points is HFS1S2F'H'.

(2) If r i is negative and r 2 positive. This type is called biconcave (fig. 9, 4). Such lenses are dispersive for all thicknesses. If d increases, the radii remaining constant, the focal lengths diminish. It is seen from the equations giving the distances of the cardinal points from the vertices that the first principal focus F is always behind SI, and the second principal focus F' always in front of S2, and that the principal points are within the lens, H' always following H. If one of the radii becomes infinite, the lens is piano-concave (fig. 9, 5).

(3) If the radii are both positive. - These lenses are called convexoconcave. Two cases occur according as r2> ri, or <ri. (a) If r 2 > r1, we obtain the mensicus (fig. 9, 3). Such lenses are always collective; and the order of the cardinal points is FHH'F'. Since sr, and sH are always negative, the object-side cardinal points are always in front of the lens. H' can take up different positions. Since sH'=-dr 2 /R= -dr i /!n(r 2 -r i) -i-d(n-I) }, sH' is greater or less than d, i.e. H' is either in front of or inside the lens, according as d <or > {r 2 -n(r 2 - r1)1/(n - I). (b) If r 2 <r i the lens is dispersive so long as d <n(ri-r2)/(n-I). H is always behind S 1 and H' behind S2, since sH and sH' are always positive. The focus F is always behind S 1 and F' in front of S2. If the thickness be small, the order of the cardinal points is F'HH'F; a dispersive lens of this type is shown in fig. 9, 6. As the thickness increases, H, H' and F move to the right, F more rapidly than H, and H more rapidly than H'; F', on the other hand, moves to the left. As with biconvex lenses, a telescopic lens, having all the cardinal points at infinity, results when d =n(ri-r2)/(n-I). If d> n(ri-r2)/(n-I), f is positive and the lens is collective. The cardinal points are in the same order as in the mensicus, viz. FHH'F'; and the relation of the principal points to the vertices is also the same as in the mensicus.

(4) If r i and r 2 are both negative. This case is reduced to (3) above, by assuming a change in the direction of the light, or, in other words, by interchanging the objectand image-spaces. The six forms shown in fig. 9 are all used in optical constructions. It may be stated fairly generally that lenses which are thicker at the middle are collective, while those which are thinnest at the middle are dispersive.

Dif f erent Positions of Object and Image

The principal points are always near the surfaces limiting the lens, and consequently the lens divides the direct pencil containing the axis into two parts. The object can be either in front of or behind Ns =Ys the lens as in fig. 10.

FIG. W. If the object point be in front of the lens, and if it be realized by rays passing from it, it is called real. If, on the other hand, the object be behind the lens, it is called virtual; it does not actually exist, and can only be realized as an image.

When we speak of " object-points," it is always understood that the rays from the object traverse the first surface of the lens before meeting the second. In the same way, images may be either real or virtual. If the image be behind the second surface, it is real, and can be intercepted on a screen. If, however, it be in front of the lens, it is visible to an eye placed behind the lens, although the rays do not actually intersect, but only appear to do so, but the l image cannot be in- 1 9 tercepted on a screen behind the lens.

Such an image is said to be virtual. These relations are shown in fig.

By referring - to the equations given above, it is seen that a thin convergent lens produces both real and virtual images of real objects, but only a real image of a virtual object, whilst a divergent lens produces a virtual image of a real object and both real and virtual images of a virtual object. The construction of a real image of a FIG. 12.

real object by a convergent lens is shown in fig. 3; and that of a virtual image of a real object by a divergent lens in fig. 12. The optical centre of a lens is a point such that, for any ray which passes through it, the incident and emergent rays are parallel. The idea of the optical centre was originally due to J. Harris (Treatise on Optics, 1 775); it is not properly a cardinal point, although it has several interesting properties. In fig. 13, let C1Pi and C 2 P 2 be two parallel radii of a biconvex lens. Join P 1 P2 and let O 1 P I and 02P2 FIG. 13.

be incident and emergent rays which have P i P 2 for the path through the lens. Then if M be the intersection of P 1 P 2 with the axis, we have angle C 1 P I M=angle C 2 P 2 M; these two angles are - for a ray travelling in the direction OiP1P202 - the angles of emergence and of incidence respectively. From the similar triangles C 2 P 2 M and C I P I M we have C 1 M: C 2 M = C 1 P I: C2P2 = r 1 :"r 2. (II) Such rays as P 1 P 2 therefore divide the distance C1C2 in the ratio of the radii, i.e. at the fixed point M, the optical centre. Calling S I M =s i, S 2 M =5 2, then C I S I =C I M-}-MS 1 =C I M-S I M, i.e. since Cisi =r 1, C I M =r i -}-s i, and similarly C 2 M =r 2 -1-s 2. Also SIS2=SIM-FMS2 =S I M-S 2 M, i.e. d=s 1 -s 2 . Then by using equation (I I) we have s 1 =r i d/(r-r 2 ) and s2=r2d/(ri-r2), and hence s i /s 2 =r i /r 2 . The vertex distances of the optical centre are therefore in the ratio of the radii.

The values of s 1 and s 2 show that the optical centre of a biconvex or biconcave lens is in the interior of the lens, that in a plano-convex or piano-concave lens it is at the vertex of the curved surface, and in a concavo-convex lens outside the lens.

The Wave-theory Derivation of the Focal Length

The formulae above have been dreived by means of geometrical rays. We here give an account of Lord Rayleigh's wave-theory derivation of the focal length of a convex lens in terms of the aperture, thickness and refractive index (Phil. Mag. 1879 (5) 8, p. 480; 1885, 20, xvi. 14 a FIG. II.

H O S, H ?'-- S 2 F P. 354); the argument is based on the principle that the optical distance from object to image is constant.

" 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 re13 h tarded from having to pass through the t hickness (d) of glass by the amount (n - i)d. The ray DAF, which tra verses only the ex treme edge of the lens, is retarded G B merely on account FIG. 14. 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 - i)d. Now if y be the semi-aperture AC of the lens, and f be the focal length CF, AF - CF (f 2 +y 2) - f = 2 y 2 /f approximately, whence f=zY2/(n - I)d. In the case of plate-glass (n - I) = z (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 semiaperture by means of d = 2y 2 (r11 -r21). 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: (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 objectand 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 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 FIG. 15.

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 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 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. S i, S2 are two centred systems. A' is a real diaphragm lying between them. B 1 and B'2 are the fittings of the systems. Then S i 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 B 1. The object-point 0 is reproduced really through the whole system in the point 0'. From the object-point 0 three diaphragms can be seen in the object-space, viz. the fitting B 1, the image of the fitting B2 and the image A of the diaphragm A' formed by the system S i. The cone of rays nearest to B2 is not received to its total extent by the fitting B 1, and the cone which has entered through B 1 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 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 S i, 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 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 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 FIG. 16.

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 threedimensional 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 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 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 (12) 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 object-reliefs (threedimensional objects), and generally an image-relief (a three-dimensional image) is conjugate to this object-relief. It is sometimes required, however, to represent by means of an optical instrument the object-relief on a plane or on a ground-glass 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 object-plane E in the object-space, perpendicular to the axis, and called the " plane focused for." All points lying in this plane are reproduced exactly on the ground-glass plane as the points 00. The circle of confusion Z on the plane focused for corresponds to the circle of confusion Z' on the ground-glass plane. The figure formed on the plane focused for by the cones of rays from all of the object-points of the total object-space 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 object-points 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 ground-glass 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 180°, 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 object-points, 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 object-relief 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 object-side 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 object-points, 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 be 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) (c) After von Rohr.

FIG. 19.

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 G After von Rohr. After von Rohr.

FIG. 20. FIG. 21.

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 image-side focus of the system. If the perspective centre lies in front of the plane focused for, then the objects Y A DE lying in front of this plane are magnified and those behind it are diminished. This is just the 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.)