Theory of Optical Representation.—If a pencil of rays,i.e.the totality of the rays proceeding from a luminous point, falls on a lens or lens system, a section of the pencil, determined by the dimensions of the system, will be transmitted. The emergent rays will have directions differing from those of the incident rays, the alteration, however, being such that the transmitted rays are convergent in the “image-point,” just as the incident rays diverge from the “object-point.” With each incident ray is associated an emergent ray; such pairs are termed “conjugate ray pairs.” Similarly we define an object-point and its image-point as “conjugate points”; all object-points lie in the “object-space,” and all image-points lie in the “image-space.”Fig. 2.The laws of optical representations were first deduced in their most general form by E. Abbe, who assumed (1) that an optical representation always exists, and (2) that to every point in the object-space there corresponds a point in the image-space, these points being mutually convertible by straight rays; in other words, with each object-point is associated one, and only one, image-point, and if the object-point be placed at the image-point, the conjugate point is the original object-point. Such a transformation is termed a “collineation,” since it transforms points into points and straight lines into straight lines. Prior to Abbe, however, James Clerk Maxwell published, in 1856, a geometrical theory of optical representation, but his methods were unknown to Abbe and to his pupils until O. Eppenstein drew attention to them. Although Maxwell’s theory is not so general as Abbe’s, it is used here since its methods permit a simple and convenient deduction of the laws.Maxwell assumed that two object-planes perpendicular to the axis are represented sharply and similarly in two image-planes also perpendicular to the axis (by “sharply” is meant that the assumed ideal instrument unites all the rays proceeding from an object-point in one of the two planes in its image-point, the rays being generally transmitted by the system). The symmetry of the axis being premised, it is sufficient to deduce laws for a plane containing the axis. In fig. 2 let O1, O2be the two points in which the perpendicular object-planes meet the axis; and since the axis corresponds to itself, the two conjugate points O′1, O′2, are at the intersections of the two image-planes with the axis. We denote the four planes by the letters O1, O2, and O′1, O′2. If two points A, C be taken in the plane O1, their images are A′, C′ in the plane O′1, and since the planes are represented similarly, we have O′1A′:O1A = O′1C′1:O1C = β1(say), in which β1is easily seen to be thelinear magnificationof the plane-pair O1, O′1. Similarly, if two points B, D be taken in the plane O2and their images B′, D′ in the plane O′2, we have O′2B′:O2B = O′2D′:O2D = β2(say), β2being the linear magnification of the plane-pair O2, O′2. The joins of A and B and of C and D intersect in a point P, and the joins of the conjugate points similarly determine the point P′.If P′ is the only possible image-point of the object-point P, then the conjugate of every ray passing through P must pass through P′. To prove this, take a third line through P intersecting the planes O1, O2in the points E, F, and by means of the magnifications β1, β2determine the conjugate points E′, F′ in the planes O′1, O′2. Since the planes O1, O2are parallel, then AC/AE = BD/BF; and since these planes are represented similarly in O′1, O′2, then A′C′/A′E′ = B′D′/B′F′. This proportion is only possible when the straight line E′F′ contains the point P′. Since P was any point whatever, it follows that every point of the object-space is represented in one and only one point in the image-space.Take a second object-point P1, vertically under P and defined by the two rays CD1, and EF1, the conjugate point P′1will be determined by the intersection of the conjugate rays C′D′1and E′F′1, the points D′1, F′1, being readily found from the magnifications β1, β2. Since PP1is parallel to CE and also to DF, then DF = D1F1. Since the plane O2is similarly represented in O′2, D′F′ = D′1F′1; this is impossible unless P′P′1be parallel to C′E′. Therefore every perpendicular object-plane is represented by a perpendicular image-plane.Let O be the intersection of the line PP1with the axis, and let O′ be its conjugate; then it may be shown that a fixed magnification β3exists for the planes O and O′. For PP1/FF1= OO1/O1O2, P′P′1/F′F′1= O′O′/O′1O′2, and F′F′1= β2FF1. Eliminating FF1and F′F′1between these ratios, we have P′P′1/PP1β2= O′O′1·O1O2/OO1. O′1O′2, or β3= β2·O′O′1·O1O2/OO1·O′1O′2,i.e.β3= β2× a product of the axial distances.The determination of the image-point of a given object-point is facilitated by means of the so-called “cardinal points” of the optical system. To determine the image-point O′1(fig. 3) corresponding to the object-point O1, we begin by choosing from the ray pencil proceeding from O1, the ray parallel with the axis,i.e.intersecting the axis at infinity. Since the axis is its own conjugate, the parallel ray through O1must intersect the axis after refraction (say at F′). Then F′ is the image-point of an object-point situated at infinity on the axis, and is termed the “second principal focus” (Germander bildseitige Brennpunkt, the image-side focus). Similarly if O′4be on the parallel through O1but in the image-space, then the conjugate ray must intersect the axis at a point (say F), which is conjugate with the point at infinity on the axis in the image-space. This point is termed the “first principal focus” (Germander objektseitige Brennpunkt, the object-side focus).Let H1, H′1be the intersections of the focal rays through F and F′ with the line O1O′4. These two points are in the position of object and image, since they are each determined by two pairs of conjugate rays (O1H1being conjugate with H′1F′, and O′4H′1with H1F). It has already been shown that object-planes perpendicular to the axis are represented by image-planes also perpendicular to the axis. Two vertical planes through H1and H′1, are related as object- and image-planes; and if these planes intersect the axis in two points H and H′, these points are named the “principal,” or “Gauss points” of the system, H being the “object-side” and H′ the “image-side principal point.” The vertical planes containing H and H′ are the “principal planes.” It is obvious that conjugate points in these planes are equidistant from the axis; in other words, the magnification β of the pair of planes is unity. An additional characteristic of the principal planes is that the object and image are direct and not inverted. The distances between F and H, and between F′ and H′ are termed the focal lengths; the former may be called the “object-side focal length” and the latter the “image-side focal length.” The two focal points and the two principal points constitute the so-called four cardinal points of the system, and with their aid the image of any object can be readily determined.Fig. 3.Equations relating to the Focal Points.—We know that the ray proceeding from the object point O1, parallel to the axis and intersecting the principal plane H in H1, passes through H′1and F′. Choose from the pencil a second ray which contains F and intersects the principal plane H in H2; then the conjugate ray must contain points corresponding to F and H2. The conjugate of F is the point at infinity on the axis,i.e.on the ray parallel to the axis. The image of H2must be in the plane H′ at the same distance from, and on the same side of, the axis, as in H′2. The straight line passing through H′2parallel to the axis intersects the ray H′1F′ in the point O′1, which must be the image of O1. If O be the foot of the perpendicular from O1to the axis, then OO1is represented by the line O′O′1also perpendicular to the axis.This construction is not applicable if the object or image be infinitely distant. For example, if the object OO1be at infinity (O being assumed to be on the axis for the sake of simplicity), so that the object appears under a constant angle w, we know that the second principal focus is conjugate with the infinitely distant axis-point. If the object is at infinity in a plane perpendicular to the axis, the image must be in the perpendicular plane through the focal point F′ (fig. 4).The size y′ of the image is readily deduced. Of the parallel rays from the object subtending the angle w, there is one which passesthrough the first principal focus F, and intersects the principal plane H in H1. Its conjugate ray passes through H′ parallel to, and at the same distance from the axis, and intersects the image-side focal plane in O′1; this point is the image of O1, and y′ is its magnitude. From the figure we have tan w = HH1/FH = y′/f, or f = y′/tan w; this equation was used by Gauss to define the focal length.Fig. 4.Referring to fig. 3, we have from the similarity of the triangles OO1F and HH2F, HH2/OO1= FH/FO, or O′O′1/OO1= FH/FO. Let y be the magnitude of the object OO1, y′ that of the image O′O′1, x the focal distance FO of the object, and f the object-side focal distance FH; then the above equation may be written y′/y = f/x. From the similar triangles H′1H′F′ and O′1O′F′, we obtain O′O′1/OO1= F′O′/F′H′. Let x′ be the focal distance of the image F′O′, and f′ the image-side focal length F′H′; then y′/y = x′/f′. The ratio of the size of the image to the size of the object is termed thelateral magnification. Denoting this by β, we haveβ = y′/y = f/x = x′/f′,(1)and alsoxx′ = ff′.(2)By differentiating equation (2) we obtaindx′= −(ff′/x2) dx or dx′/dx = −ff′/x2.(3)The ratio of the displacement of the image dx′ to the displacement of the object dx is the axial magnification, and is denoted by α. Equation (3) gives important information on the displacement of the image when the object is moved. Since f and f′ always have contrary signs (as is proved below), the product −ff′ is invariably positive, and since x2is positive for all values of x, it follows that dx and dx′ have the same sign,i.e.the object and image always move in the same direction, either both in the direction of the light, or both in the opposite direction. This is shown in fig. 3 by the object O3O2and the image O′3O′2.If two conjugate rays be drawn from two conjugate points on the axis, making angles u and u′ with the axis, as for example the rays OH1, O′H′1, in fig. 3, u is termed the “angular aperture for the object,” and u′ the “angular aperture for the image.” The ratio of the tangents of these angles is termed the “convergence” and is denoted by γ, thus γ = tan u′/tan u. Now tan u′= H′H′1/O′H′ = H′H′1/(O′F′ + F′H′) = H′H′1/(F′H′ − F′O′). Also tan u = HH1/OH = HH1/(OF + FH) = HH1/(FH − FO). Consequently γ = (FH − FO)/(F′H′ − F′O′), or, in our previous notation, γ = (f − x)/(f′ − x′).From equation (1) f/x = x′/f′, we obtain by subtracting unity from both sides (f − x)/x = (x′ − f′)/f′, and consequentlyf − x= −x= −f= γ.f′ − x′f′x′(4)From equations (1), (3) and (4), it is seen that a simple relation exists between the lateral magnification, the axial magnification and the convergence, viz. αγ = β.Fig. 5.In addition to the four cardinal points F, H, F′, H′, J. B. Listing, “Beiträge aus physiologischen Optik,”Göttinger Studien(1845) introduced the so-called “nodal points” (Knotenpunkte) of the system, which are the two conjugate points from which the object and image appear under the same angle. In fig. 5 let K be the nodal point from which the object y appears under the same angle as the image y′ from the other nodal point K′. Then OO1/KO = O′O′1/K′O′, or OO1/(KF + FO) = O′O′1/(K′F′+ F′O′), or OO1/(FO − FK) = O′O′1/(F′O′− F′K′). Calling the focal distances FK and F′K′, X and X′, we have y/(x − X) = y′/(x′− X′), and since y′/y = β, it follows that 1/(x − X) = β/(x′− X′). Replace x′ and X′ by the values given in equation (2), and we obtain1= β /(ff′−ff′)or 1 = −βxX.x − XxXff′Since β = f/x = x′/f′, we have f′ = −X, f = −X′.These equations show that to determine the nodal points, it is only necessary to measure the focal distance of the second principal focus from the first principal focus, and vice versa. In the special case when the initial and final medium is the same, as for example, a lens in air, we have f = −f′, and the nodal points coincide with the principal points of the system; we then speak of the “nodal point property of the principal points,” meaning that the object and corresponding image subtend the same angle at the principal points.Equations Relating to the Principal Points.—It is sometimes desirable to determine the distances of an object and its image, not from the focal points, but from the principal points. Let A (see fig. 3) be the principal point distance of the object and A′ that of the image, we then haveA = HO = HF + FO = FO − FH = x − f,A′ = H′O′ = H′F′ + F′O′ = F′O′ − F′H′ = x′ − f′,whencex = A + f and x′ = A′ + f′.Using xx′ = ff′, we have (A + f)(A′ + f′) = ff′, which leads to AA′ + Af′ + A′f = O, or1 +f′+f= O;A′Athis becomes in the special case when f = -f′,1−1=1.A′AfTo express the linear magnification in terms of the principal point distances, we start with equation (4) (f − x)/(f′ − x′) = −x/f′. From this we obtain A/A′ = -x/f′, or x = −f′A/A′; and by using equation (1) we have β = −fA′/f′A.In the special case of f = −f′, this becomes β = A′/A = y′/y, from which it follows that the ratio of the dimensions of the object and image is equal to the ratio of the distances of the object and image from the principal points.The convergence can be determined in terms of A and A′ by substituting x = −f′A/A′ in equation (4), when we obtain γ = A/A′.Compound Systems.—In discussing the laws relating to compound systems, we assume that the cardinal points of the component systems are known, and also that the combinations are centred,i.e.that the axes of the component lenses coincide. If some object be represented by two systems arranged one behind the other, we can regard the systems as co-operating in the formation of the final image.Fig. 6.Let such a system be represented in fig. 6. The two single systems are denoted by the suffixes 1 and 2; for example, F1is the first principal focus of the first, and F′2the second principal focus of the second system. A ray parallel to the axis at a distance y passes through the second principal focus F′1of the first system, intersecting the axis at an angle w′1. The point F′1will be represented in the second system by the point F′, which is therefore conjugate to the point at infinity for the entire system,i.e.it is the second principal focus of the compound system. The representation of F′1in F′ by the second system leads to the relations F2F′1= x2, and F′2F′ = x′2, whence x2x′2= f2f′2. Denoting the distance between the adjacent focal planes F′1, F2by Δ, we have Δ = F′1F2= −F2F′1, so that x′2= -f2f′2/Δ. A similar ray parallel to the axis at a distance y proceeding from the image-side will intersect the axis at the focal point F2; and by finding the image of this point in the first system, we determine the first principal focus of the compound system. Equation (2) gives x1x′1= f1f′1, and since x′1= F′1F2= Δ, we have x1= f1f′1/Δ as the distance of the first principal focus F of the compound system from the first principal focus F1of the first system.To determine the focal lengths f and f′ of the compound system and the principal points H and H′, we employ the equations defining the focal lengths, viz. f = y′/tan w, and f′ = y/tan w′. From the construction (fig. 6) tan w′1= y/f′1. The variation of the angle w′1by the second system is deduced from the equation to the convergence, viz. γ = tan w′2/tan w2= −x2/f′2= Δ/f′2, and since w2= w′1, we have tan w′2= (Δ/f′2) tan w′1. Since w′ = w′2in our system of notation, we havef′ =y=yf′2=f′1·f′2.tan w′Δ tan w′1ΔBy taking a ray proceeding from the image-side we obtain for the first principal focal distance of the combinationf = −f1f2/Δ.In the particular case in which Δ = 0, the two focal planes F′1, F2coincide, and the focal lengths f, f′ are infinite. Such a system is called a telescopic system, and this condition is realized in a telescope focused for a normal eye.So far we have assumed that all the rays proceeding from an object-point are exactly united in an image-point after transmission through the ideal system. The question now arisesasto how far this assumption is justified for spherical lenses. To investigate this it is simplest to trace the path of a ray through one sphericalrefracting surface. Let such a surface divide media of refractive indices n and n′, the former being to the left. The point where the axis intersects the surface is the vertex S (fig. 7). Denote the distance of the axial object-point O from S by s; the distance from O to the point of incidence P by p; the radius of the spherical surface by r; and the distance OC by c, C being the centre of the sphere. Let u be the angle made by the ray with the axis, and i the angle of incidence,i.e.the angle between the ray and the normal to the sphere at the point of incidence. The corresponding quantities in the image-space are denoted by the same letters with a dash. From the triangle O′PC we have sin u = (r/c) sin i, and from the triangle O′PC we have sin u′ = (r/c′) sin i′. By Snell’s law we have n′/n = sin i/sin i′, and also φ = u′ + i′. Consequently c′ and the position of the image may be found.Fig. 7.To determine whether all the rays proceeding from O are refracted through O′, we investigate the triangle OPO′. We have p/p′ = sin u′/sin u. Substituting for sin u and sin u′ the values found above, we obtain p′/p = c′ sin i/c sin i′ = n′c′/nc. Also c = OC = CS + SO = −SC + SO = s − r, and similarly c′ = s′ − r. Substituting these values we obtainp′=n′(s′ − r), orn(s − r)=n′(s′ − r).pn(s − r)pp′(6)To obtain p and p′ we use the triangles OPC and O′PC; we have p2= (s − r)2+ r2+ 2r(s − r) cos φ, p′2= (s′ − r)2+ r2+ 2r(s′ − r) cos φ. Hence if s, r, n and n′ be constant, s′ must vary as φ varies. The refracted rays therefore do not reunite in a point, and the deflection is termed the spherical aberration (seeAberration).Developing cos φ in powers of φ, we obtainp2= (s − r)2+ r2+ 2r(s − r){1 −φ2+φ4−φ6+ ...},2!4!6!and therefore for such values of φ for which the second and higher powers may be neglected, we have p2= (s − r)2+ r2+ 2r(s − r),i.e.p = s, and similarly p′ = s′. Equation (6) then becomes n(s − r)/s = n′(s′ − r)/s′ orn′=n+n′ − n.s′sr(7)This relation shows that in a very small central aperture in which the equation p = s holds, all rays proceeding from an object-point are exactly united in an image-point, and therefore the equations previously deduced are valid for this aperture. K. F. Gauss derived the equations for thin pencils in hisDioptrische Untersuchungen(1840) by very elegant methods. More recently the laws relating to systems with finite aperture have been approximately realized, as for example, in well-corrected photographic objectives.Position of the Cardinal Points of a Lens.—Taking the case of a single spherical refracting surface, and limiting ourselves to the small central aperture, it is seen that the second principal focus F′ is obtained when s is infinitely great. Consequently s′ = -f′; the difference of sign is obvious, since s′ is measured from S, while f′ is measured from F′. The focal lengths are directly deducible from equation (7):—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 r1, r2of 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 F1be the first principal focus of the first system of radius r1, and F1′ the second principal focus; and let S1be its vertex. Denote the distance F1S1(the first principal focal length) by f1, and the corresponding distance F′1S1by f′1. Let the corresponding quantities in the second system be denoted by the same letters with the suffix 2.By equations (8) and (9) we havef1=r1, f′1= −nr1, f2= −nr2, f′2=r2,n − 1n − 1n − 1n − 1f2having the opposite sign to f1. Denoting the distance F′1F2by Δ, we have Δ = F′1F2= F′1S1+ S1S2+ S2F2= F′1S1+ S1S2− F2S2= f′1+ d − f2. Substituting for f′1and f2we obtainΔ = −nr1+ d +nr2.n − 1n − 1Writing R = Δ(n − 1), this relation becomesR = n(r2− r1) + d(n − 1).We have already shown that f (the first principal focal length of a compound system) = −f1f2/Δ. Substituting for f1, f2and Δ the values found above, we obtainf =r1r2n=r1r2n,(n − 1)R(n − 1) {n (r2− r1) + d(n − 1)}(10)which is equivalent to1= (n − 1){1−1}+(n − 1)2d.fr1r2r1r2nIf the lens be infinitely thin,i.e.if d be zero, we have for the first principal focal length.1= (n − 1){1−1}.fr1r2By the same method we obtain for the second principal focal lengthf′ =f′1f′2= −nr1r2= −f.Δ(n − 1)RFig. 8.The reciprocal of the focal length is termed thepowerof the lens and is denoted by φ. In formulae involving φ it is customary to denote the reciprocal of the radii by the symbol ρ; we thus have φ = 1/f, ρ = 1/r. Equation (10) thus becomesφ = (n − 1) (ρ1− ρ2) +(n − 1)2dρ1ρ2.nThe unit of power employed by spectacle-makers is termed thediopterordioptric(seeSpectacles).We proceed to determine the distances of the focal points from the vertices of the lens,i.e.the distances FS1and F′S2. Since F is represented by the first system in F2, we have by equation (2)x1=f1f′1=f1f′1= −nr12,x′1Δ(n − 1)Rwhere x1= F1F, and x′1= F′1F2= Δ. The distance of the first principal focus from the vertex S,i.e.S1F, which we denote by sFis given by sF= S1F = S1F1+ F1F = −F1S1+ F1F. Now F1S1is the distance from the vertex of the first principal focus of the first system,i.e.f1and F1F = x1. Substituting these values, we obtainsF= −r1−nr12= −r1(nr1+ R).n − 1(n − 1)R(n − 1)RThe distance F′2F′ or x′2is similarly determined by considering F′1to be represented by the second system in F′.We havex′2=f2f′2= −f2f′2=nr22,x2Δ(n − 1)Rso thatsF′ = x′2− f′2=r2(nr2− R),(n − 1)Rwhere sF′ denotes the distance of the second principal focus from the vertex S2.The two focal lengths and the distances of the foci from the vertices being known, the positions of the remaining cardinal points,i.e.the principal points H and H′, are readily determined. Let sH= S1H,i.e.the distance of the object-side principal point from the vertex of the first surface, and sH′= S2H′,i.e.the distance of the image-side principal point from the vertex of the second surface, then f = FH = FS1+ S1H = −S1F + S1H = −sF+ sH; hence sH= sF+ f = −dr1/R. Similarly sH′= sF′+ f′ = −dr2/R. It is readily seen that the distances sHand sH′are in the ratio of the radii r1and r2.The distance between the two principal planes (the interstitium) is deduced very simply. We have S1S2= S1H + HH′ + H′S2, or HH′ = S1S2− S1H + S2H′. Substituting, we haveHH′ = d − sH+ sH′= d(n − 1) (r2− r1+ d)/R.The interstitium becomes zero, or the two principal planes coincide, if d = r1− r2.We have now derived all the properties of the lens in terms of its elements, viz. the refractive index, the radii of the surfaces, and the thickness.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 collecting, convergentand 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 (10).f =r1r2n.(n − 1) { n (r2− r1) + d(n − 1)}Fig. 9.(1) If r1be positive and r2negative. This type is called biconvex (fig. 9, 1). The first principal focus is in front of the lens, and the second principal focus behind the lens, and the two principal points are inside the lens. The order of the cardinal points is therefore FS1HH′S2F′. The lens is convergent so long as the thickness is less than n(r1− r2)/(n − 1). 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 termedplano-convex. As d increases, F and H move to the right and F′ and H′ to the left. If d = n(r1− r2)/(n − 1), the focal length is infinite,i.e.the lens is telescopic. If the thickness be greater than n(r1− r2)/(n − 1), the lens is dispersive, and the order of the cardinal points is HFS1S2F′H′.(2) If r1is negative and r2positive. This type is calledbiconcave(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 S1, 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 isplano-concave(fig. 9, 5).(3) If the radii are both positive. These lenses are calledconvexo-concave. Two cases occur according as r2> r1, or < r1. (a) If r2> r1, we obtain themensicus(fig. 9, 3). Such lenses are always collective; and the order of the cardinal points is FHH′F′. Since sFand sHare always negative, the object-side cardinal points are always in front of the lens. H′ can take up different positions. Since sH′= −dr2/R = −dr2/{n (r2− r1) + d(n − 1)}, sH′is greater or less than d,i.e.H′ is either in front of or inside the lens, according as d < or > {r2− n(r2− r1)}/(n − 1). (b) If r2< r1the lens is dispersive so long as d < n(r1− r2)/(n − 1). H is always behind S1and H′ behind S2, since sHand sH′are always positive. The focus F is always behind S1and 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(r1− r2)/(n − 1). If d > n(r1− r2)/(n − 1), 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 r1and r2are 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 object- and 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.Fig. 10.Different 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 the lens as in fig. 10. If the object point be in front of the lens, and if it be realized by rays passing from it, it is calledreal. If, on the other hand, the object be behind the lens, it is calledvirtual; it does not actually exist, and can only be realized as an image.Fig. 11.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 isreal, 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 image cannot be intercepted on a screen behind the lens. Such an image is said to bevirtual. These relations are shown in fig. 11.Fig. 12.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 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.Fig. 13.The optical centre of a lensis 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, 1775); it is not properly a cardinal point, although it has several interesting properties. In fig. 13, let C1P1and C2P2be two parallel radii of a biconvex lens. Join P1P2and let O1P1and O2P2be incident and emergent rays which have P1P2for the path through the lens. Then if M be the intersection of P1P2with the axis, we have angle C1P1M = angle C2P2M; these two angles are—for a ray travelling in the direction O1P1P2O2—the angles of emergence and of incidence respectively. From the similar triangles C2P2M and C1P1M we haveC1M : C2M = C1P1: C2P2= r1: r2.(11)Such rays as P1P2therefore divide the distance C1C2in the ratio of the radii,i.e.at the fixed point M, the optical centre. Calling S1M = s1, S2M = s2, then C1S1= C1M + MS1= C1M − S1M,i.e.since C1S1= r1, C1M = r1+ s1, and similarly C2M = r2+ s2. Also S1S2= S1M + MS2= S1M − S2M,i.e.d = s1− s2. Then by using equation (11) we have s1= r1d/(r − r2) and s2= r2d/(r1− r2), and hence s1/s2= r1/r2. The vertex distances of the optical centre are therefore in the ratio of the radii.The values of s1and s2show that the optical centre of a biconvex or biconcave lens is in the interior of the lens, that in a plano-convex or plano-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 derived 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,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 = ½y2/f approximately, whencef = ½y2/ (n − 1)d.(12)In the case of plate-glass (n − 1) = ½ (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 = ½y2(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.”
Theory of Optical Representation.—If a pencil of rays,i.e.the totality of the rays proceeding from a luminous point, falls on a lens or lens system, a section of the pencil, determined by the dimensions of the system, will be transmitted. The emergent rays will have directions differing from those of the incident rays, the alteration, however, being such that the transmitted rays are convergent in the “image-point,” just as the incident rays diverge from the “object-point.” With each incident ray is associated an emergent ray; such pairs are termed “conjugate ray pairs.” Similarly we define an object-point and its image-point as “conjugate points”; all object-points lie in the “object-space,” and all image-points lie in the “image-space.”
The laws of optical representations were first deduced in their most general form by E. Abbe, who assumed (1) that an optical representation always exists, and (2) that to every point in the object-space there corresponds a point in the image-space, these points being mutually convertible by straight rays; in other words, with each object-point is associated one, and only one, image-point, and if the object-point be placed at the image-point, the conjugate point is the original object-point. Such a transformation is termed a “collineation,” since it transforms points into points and straight lines into straight lines. Prior to Abbe, however, James Clerk Maxwell published, in 1856, a geometrical theory of optical representation, but his methods were unknown to Abbe and to his pupils until O. Eppenstein drew attention to them. Although Maxwell’s theory is not so general as Abbe’s, it is used here since its methods permit a simple and convenient deduction of the laws.
Maxwell assumed that two object-planes perpendicular to the axis are represented sharply and similarly in two image-planes also perpendicular to the axis (by “sharply” is meant that the assumed ideal instrument unites all the rays proceeding from an object-point in one of the two planes in its image-point, the rays being generally transmitted by the system). The symmetry of the axis being premised, it is sufficient to deduce laws for a plane containing the axis. In fig. 2 let O1, O2be the two points in which the perpendicular object-planes meet the axis; and since the axis corresponds to itself, the two conjugate points O′1, O′2, are at the intersections of the two image-planes with the axis. We denote the four planes by the letters O1, O2, and O′1, O′2. If two points A, C be taken in the plane O1, their images are A′, C′ in the plane O′1, and since the planes are represented similarly, we have O′1A′:O1A = O′1C′1:O1C = β1(say), in which β1is easily seen to be thelinear magnificationof the plane-pair O1, O′1. Similarly, if two points B, D be taken in the plane O2and their images B′, D′ in the plane O′2, we have O′2B′:O2B = O′2D′:O2D = β2(say), β2being the linear magnification of the plane-pair O2, O′2. The joins of A and B and of C and D intersect in a point P, and the joins of the conjugate points similarly determine the point P′.
If P′ is the only possible image-point of the object-point P, then the conjugate of every ray passing through P must pass through P′. To prove this, take a third line through P intersecting the planes O1, O2in the points E, F, and by means of the magnifications β1, β2determine the conjugate points E′, F′ in the planes O′1, O′2. Since the planes O1, O2are parallel, then AC/AE = BD/BF; and since these planes are represented similarly in O′1, O′2, then A′C′/A′E′ = B′D′/B′F′. This proportion is only possible when the straight line E′F′ contains the point P′. Since P was any point whatever, it follows that every point of the object-space is represented in one and only one point in the image-space.
Take a second object-point P1, vertically under P and defined by the two rays CD1, and EF1, the conjugate point P′1will be determined by the intersection of the conjugate rays C′D′1and E′F′1, the points D′1, F′1, being readily found from the magnifications β1, β2. Since PP1is parallel to CE and also to DF, then DF = D1F1. Since the plane O2is similarly represented in O′2, D′F′ = D′1F′1; this is impossible unless P′P′1be parallel to C′E′. Therefore every perpendicular object-plane is represented by a perpendicular image-plane.
Let O be the intersection of the line PP1with the axis, and let O′ be its conjugate; then it may be shown that a fixed magnification β3exists for the planes O and O′. For PP1/FF1= OO1/O1O2, P′P′1/F′F′1= O′O′/O′1O′2, and F′F′1= β2FF1. Eliminating FF1and F′F′1between these ratios, we have P′P′1/PP1β2= O′O′1·O1O2/OO1. O′1O′2, or β3= β2·O′O′1·O1O2/OO1·O′1O′2,i.e.β3= β2× a product of the axial distances.
The determination of the image-point of a given object-point is facilitated by means of the so-called “cardinal points” of the optical system. To determine the image-point O′1(fig. 3) corresponding to the object-point O1, we begin by choosing from the ray pencil proceeding from O1, the ray parallel with the axis,i.e.intersecting the axis at infinity. Since the axis is its own conjugate, the parallel ray through O1must intersect the axis after refraction (say at F′). Then F′ is the image-point of an object-point situated at infinity on the axis, and is termed the “second principal focus” (Germander bildseitige Brennpunkt, the image-side focus). Similarly if O′4be on the parallel through O1but in the image-space, then the conjugate ray must intersect the axis at a point (say F), which is conjugate with the point at infinity on the axis in the image-space. This point is termed the “first principal focus” (Germander objektseitige Brennpunkt, the object-side focus).
Let H1, H′1be the intersections of the focal rays through F and F′ with the line O1O′4. These two points are in the position of object and image, since they are each determined by two pairs of conjugate rays (O1H1being conjugate with H′1F′, and O′4H′1with H1F). It has already been shown that object-planes perpendicular to the axis are represented by image-planes also perpendicular to the axis. Two vertical planes through H1and H′1, are related as object- and image-planes; and if these planes intersect the axis in two points H and H′, these points are named the “principal,” or “Gauss points” of the system, H being the “object-side” and H′ the “image-side principal point.” The vertical planes containing H and H′ are the “principal planes.” It is obvious that conjugate points in these planes are equidistant from the axis; in other words, the magnification β of the pair of planes is unity. An additional characteristic of the principal planes is that the object and image are direct and not inverted. The distances between F and H, and between F′ and H′ are termed the focal lengths; the former may be called the “object-side focal length” and the latter the “image-side focal length.” The two focal points and the two principal points constitute the so-called four cardinal points of the system, and with their aid the image of any object can be readily determined.
Equations relating to the Focal Points.—We know that the ray proceeding from the object point O1, parallel to the axis and intersecting the principal plane H in H1, passes through H′1and F′. Choose from the pencil a second ray which contains F and intersects the principal plane H in H2; then the conjugate ray must contain points corresponding to F and H2. The conjugate of F is the point at infinity on the axis,i.e.on the ray parallel to the axis. The image of H2must be in the plane H′ at the same distance from, and on the same side of, the axis, as in H′2. The straight line passing through H′2parallel to the axis intersects the ray H′1F′ in the point O′1, which must be the image of O1. If O be the foot of the perpendicular from O1to the axis, then OO1is represented by the line O′O′1also perpendicular to the axis.
This construction is not applicable if the object or image be infinitely distant. For example, if the object OO1be at infinity (O being assumed to be on the axis for the sake of simplicity), so that the object appears under a constant angle w, we know that the second principal focus is conjugate with the infinitely distant axis-point. If the object is at infinity in a plane perpendicular to the axis, the image must be in the perpendicular plane through the focal point F′ (fig. 4).
The size y′ of the image is readily deduced. Of the parallel rays from the object subtending the angle w, there is one which passesthrough the first principal focus F, and intersects the principal plane H in H1. Its conjugate ray passes through H′ parallel to, and at the same distance from the axis, and intersects the image-side focal plane in O′1; this point is the image of O1, and y′ is its magnitude. From the figure we have tan w = HH1/FH = y′/f, or f = y′/tan w; this equation was used by Gauss to define the focal length.
Referring to fig. 3, we have from the similarity of the triangles OO1F and HH2F, HH2/OO1= FH/FO, or O′O′1/OO1= FH/FO. Let y be the magnitude of the object OO1, y′ that of the image O′O′1, x the focal distance FO of the object, and f the object-side focal distance FH; then the above equation may be written y′/y = f/x. From the similar triangles H′1H′F′ and O′1O′F′, we obtain O′O′1/OO1= F′O′/F′H′. Let x′ be the focal distance of the image F′O′, and f′ the image-side focal length F′H′; then y′/y = x′/f′. The ratio of the size of the image to the size of the object is termed thelateral magnification. Denoting this by β, we have
β = y′/y = f/x = x′/f′,
(1)
and also
xx′ = ff′.
(2)
By differentiating equation (2) we obtain
dx′= −(ff′/x2) dx or dx′/dx = −ff′/x2.
(3)
The ratio of the displacement of the image dx′ to the displacement of the object dx is the axial magnification, and is denoted by α. Equation (3) gives important information on the displacement of the image when the object is moved. Since f and f′ always have contrary signs (as is proved below), the product −ff′ is invariably positive, and since x2is positive for all values of x, it follows that dx and dx′ have the same sign,i.e.the object and image always move in the same direction, either both in the direction of the light, or both in the opposite direction. This is shown in fig. 3 by the object O3O2and the image O′3O′2.
If two conjugate rays be drawn from two conjugate points on the axis, making angles u and u′ with the axis, as for example the rays OH1, O′H′1, in fig. 3, u is termed the “angular aperture for the object,” and u′ the “angular aperture for the image.” The ratio of the tangents of these angles is termed the “convergence” and is denoted by γ, thus γ = tan u′/tan u. Now tan u′= H′H′1/O′H′ = H′H′1/(O′F′ + F′H′) = H′H′1/(F′H′ − F′O′). Also tan u = HH1/OH = HH1/(OF + FH) = HH1/(FH − FO). Consequently γ = (FH − FO)/(F′H′ − F′O′), or, in our previous notation, γ = (f − x)/(f′ − x′).
From equation (1) f/x = x′/f′, we obtain by subtracting unity from both sides (f − x)/x = (x′ − f′)/f′, and consequently
(4)
From equations (1), (3) and (4), it is seen that a simple relation exists between the lateral magnification, the axial magnification and the convergence, viz. αγ = β.
In addition to the four cardinal points F, H, F′, H′, J. B. Listing, “Beiträge aus physiologischen Optik,”Göttinger Studien(1845) introduced the so-called “nodal points” (Knotenpunkte) of the system, which are the two conjugate points from which the object and image appear under the same angle. In fig. 5 let K be the nodal point from which the object y appears under the same angle as the image y′ from the other nodal point K′. Then OO1/KO = O′O′1/K′O′, or OO1/(KF + FO) = O′O′1/(K′F′+ F′O′), or OO1/(FO − FK) = O′O′1/(F′O′− F′K′). Calling the focal distances FK and F′K′, X and X′, we have y/(x − X) = y′/(x′− X′), and since y′/y = β, it follows that 1/(x − X) = β/(x′− X′). Replace x′ and X′ by the values given in equation (2), and we obtain
Since β = f/x = x′/f′, we have f′ = −X, f = −X′.
These equations show that to determine the nodal points, it is only necessary to measure the focal distance of the second principal focus from the first principal focus, and vice versa. In the special case when the initial and final medium is the same, as for example, a lens in air, we have f = −f′, and the nodal points coincide with the principal points of the system; we then speak of the “nodal point property of the principal points,” meaning that the object and corresponding image subtend the same angle at the principal points.
Equations Relating to the Principal Points.—It is sometimes desirable to determine the distances of an object and its image, not from the focal points, but from the principal points. Let A (see fig. 3) be the principal point distance of the object and A′ that of the image, we then have
A = HO = HF + FO = FO − FH = x − f,A′ = H′O′ = H′F′ + F′O′ = F′O′ − F′H′ = x′ − f′,
whence
x = A + f and x′ = A′ + f′.
Using xx′ = ff′, we have (A + f)(A′ + f′) = ff′, which leads to AA′ + Af′ + A′f = O, or
this becomes in the special case when f = -f′,
To express the linear magnification in terms of the principal point distances, we start with equation (4) (f − x)/(f′ − x′) = −x/f′. From this we obtain A/A′ = -x/f′, or x = −f′A/A′; and by using equation (1) we have β = −fA′/f′A.
In the special case of f = −f′, this becomes β = A′/A = y′/y, from which it follows that the ratio of the dimensions of the object and image is equal to the ratio of the distances of the object and image from the principal points.
The convergence can be determined in terms of A and A′ by substituting x = −f′A/A′ in equation (4), when we obtain γ = A/A′.
Compound Systems.—In discussing the laws relating to compound systems, we assume that the cardinal points of the component systems are known, and also that the combinations are centred,i.e.that the axes of the component lenses coincide. If some object be represented by two systems arranged one behind the other, we can regard the systems as co-operating in the formation of the final image.
Let such a system be represented in fig. 6. The two single systems are denoted by the suffixes 1 and 2; for example, F1is the first principal focus of the first, and F′2the second principal focus of the second system. A ray parallel to the axis at a distance y passes through the second principal focus F′1of the first system, intersecting the axis at an angle w′1. The point F′1will be represented in the second system by the point F′, which is therefore conjugate to the point at infinity for the entire system,i.e.it is the second principal focus of the compound system. The representation of F′1in F′ by the second system leads to the relations F2F′1= x2, and F′2F′ = x′2, whence x2x′2= f2f′2. Denoting the distance between the adjacent focal planes F′1, F2by Δ, we have Δ = F′1F2= −F2F′1, so that x′2= -f2f′2/Δ. A similar ray parallel to the axis at a distance y proceeding from the image-side will intersect the axis at the focal point F2; and by finding the image of this point in the first system, we determine the first principal focus of the compound system. Equation (2) gives x1x′1= f1f′1, and since x′1= F′1F2= Δ, we have x1= f1f′1/Δ as the distance of the first principal focus F of the compound system from the first principal focus F1of the first system.
To determine the focal lengths f and f′ of the compound system and the principal points H and H′, we employ the equations defining the focal lengths, viz. f = y′/tan w, and f′ = y/tan w′. From the construction (fig. 6) tan w′1= y/f′1. The variation of the angle w′1by the second system is deduced from the equation to the convergence, viz. γ = tan w′2/tan w2= −x2/f′2= Δ/f′2, and since w2= w′1, we have tan w′2= (Δ/f′2) tan w′1. Since w′ = w′2in our system of notation, we have
By taking a ray proceeding from the image-side we obtain for the first principal focal distance of the combination
f = −f1f2/Δ.
In the particular case in which Δ = 0, the two focal planes F′1, F2coincide, and the focal lengths f, f′ are infinite. Such a system is called a telescopic system, and this condition is realized in a telescope focused for a normal eye.
So far we have assumed that all the rays proceeding from an object-point are exactly united in an image-point after transmission through the ideal system. The question now arisesasto how far this assumption is justified for spherical lenses. To investigate this it is simplest to trace the path of a ray through one sphericalrefracting surface. Let such a surface divide media of refractive indices n and n′, the former being to the left. The point where the axis intersects the surface is the vertex S (fig. 7). Denote the distance of the axial object-point O from S by s; the distance from O to the point of incidence P by p; the radius of the spherical surface by r; and the distance OC by c, C being the centre of the sphere. Let u be the angle made by the ray with the axis, and i the angle of incidence,i.e.the angle between the ray and the normal to the sphere at the point of incidence. The corresponding quantities in the image-space are denoted by the same letters with a dash. From the triangle O′PC we have sin u = (r/c) sin i, and from the triangle O′PC we have sin u′ = (r/c′) sin i′. By Snell’s law we have n′/n = sin i/sin i′, and also φ = u′ + i′. Consequently c′ and the position of the image may be found.
To determine whether all the rays proceeding from O are refracted through O′, we investigate the triangle OPO′. We have p/p′ = sin u′/sin u. Substituting for sin u and sin u′ the values found above, we obtain p′/p = c′ sin i/c sin i′ = n′c′/nc. Also c = OC = CS + SO = −SC + SO = s − r, and similarly c′ = s′ − r. Substituting these values we obtain
(6)
To obtain p and p′ we use the triangles OPC and O′PC; we have p2= (s − r)2+ r2+ 2r(s − r) cos φ, p′2= (s′ − r)2+ r2+ 2r(s′ − r) cos φ. Hence if s, r, n and n′ be constant, s′ must vary as φ varies. The refracted rays therefore do not reunite in a point, and the deflection is termed the spherical aberration (seeAberration).
Developing cos φ in powers of φ, we obtain
and therefore for such values of φ for which the second and higher powers may be neglected, we have p2= (s − r)2+ r2+ 2r(s − r),i.e.p = s, and similarly p′ = s′. Equation (6) then becomes n(s − r)/s = n′(s′ − r)/s′ or
(7)
This relation shows that in a very small central aperture in which the equation p = s holds, all rays proceeding from an object-point are exactly united in an image-point, and therefore the equations previously deduced are valid for this aperture. K. F. Gauss derived the equations for thin pencils in hisDioptrische Untersuchungen(1840) by very elegant methods. More recently the laws relating to systems with finite aperture have been approximately realized, as for example, in well-corrected photographic objectives.
Position of the Cardinal Points of a Lens.—Taking the case of a single spherical refracting surface, and limiting ourselves to the small central aperture, it is seen that the second principal focus F′ is obtained when s is infinitely great. Consequently s′ = -f′; the difference of sign is obvious, since s′ is measured from S, while f′ is measured from F′. The focal lengths are directly deducible from equation (7):—
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 r1, r2of 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 F1be the first principal focus of the first system of radius r1, and F1′ the second principal focus; and let S1be its vertex. Denote the distance F1S1(the first principal focal length) by f1, and the corresponding distance F′1S1by f′1. Let the corresponding quantities in the second system be denoted by the same letters with the suffix 2.
By equations (8) and (9) we have
f2having the opposite sign to f1. Denoting the distance F′1F2by Δ, we have Δ = F′1F2= F′1S1+ S1S2+ S2F2= F′1S1+ S1S2− F2S2= f′1+ d − f2. Substituting for f′1and f2we obtain
Writing R = Δ(n − 1), this relation becomes
R = n(r2− r1) + d(n − 1).
We have already shown that f (the first principal focal length of a compound system) = −f1f2/Δ. Substituting for f1, f2and Δ the values found above, we obtain
(10)
which is equivalent to
If the lens be infinitely thin,i.e.if d be zero, we have for the first principal focal length.
By the same method we obtain for the second principal focal length
The reciprocal of the focal length is termed thepowerof the lens and is denoted by φ. In formulae involving φ it is customary to denote the reciprocal of the radii by the symbol ρ; we thus have φ = 1/f, ρ = 1/r. Equation (10) thus becomes
The unit of power employed by spectacle-makers is termed thediopterordioptric(seeSpectacles).
We proceed to determine the distances of the focal points from the vertices of the lens,i.e.the distances FS1and F′S2. Since F is represented by the first system in F2, we have by equation (2)
where x1= F1F, and x′1= F′1F2= Δ. The distance of the first principal focus from the vertex S,i.e.S1F, which we denote by sFis given by sF= S1F = S1F1+ F1F = −F1S1+ F1F. Now F1S1is the distance from the vertex of the first principal focus of the first system,i.e.f1and F1F = x1. Substituting these values, we obtain
The distance F′2F′ or x′2is similarly determined by considering F′1to be represented by the second system in F′.
We have
so that
where sF′ denotes the distance of the second principal focus from the vertex S2.
The two focal lengths and the distances of the foci from the vertices being known, the positions of the remaining cardinal points,i.e.the principal points H and H′, are readily determined. Let sH= S1H,i.e.the distance of the object-side principal point from the vertex of the first surface, and sH′= S2H′,i.e.the distance of the image-side principal point from the vertex of the second surface, then f = FH = FS1+ S1H = −S1F + S1H = −sF+ sH; hence sH= sF+ f = −dr1/R. Similarly sH′= sF′+ f′ = −dr2/R. It is readily seen that the distances sHand sH′are in the ratio of the radii r1and r2.
The distance between the two principal planes (the interstitium) is deduced very simply. We have S1S2= S1H + HH′ + H′S2, or HH′ = S1S2− S1H + S2H′. Substituting, we have
HH′ = d − sH+ sH′= d(n − 1) (r2− r1+ d)/R.
The interstitium becomes zero, or the two principal planes coincide, if d = r1− r2.
We have now derived all the properties of the lens in terms of its elements, viz. the refractive index, the radii of the surfaces, and the thickness.
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 collecting, convergentand 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 (10).
(1) If r1be positive and r2negative. This type is called biconvex (fig. 9, 1). The first principal focus is in front of the lens, and the second principal focus behind the lens, and the two principal points are inside the lens. The order of the cardinal points is therefore FS1HH′S2F′. The lens is convergent so long as the thickness is less than n(r1− r2)/(n − 1). 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 termedplano-convex. As d increases, F and H move to the right and F′ and H′ to the left. If d = n(r1− r2)/(n − 1), the focal length is infinite,i.e.the lens is telescopic. If the thickness be greater than n(r1− r2)/(n − 1), the lens is dispersive, and the order of the cardinal points is HFS1S2F′H′.
(2) If r1is negative and r2positive. This type is calledbiconcave(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 S1, 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 isplano-concave(fig. 9, 5).
(3) If the radii are both positive. These lenses are calledconvexo-concave. Two cases occur according as r2> r1, or < r1. (a) If r2> r1, we obtain themensicus(fig. 9, 3). Such lenses are always collective; and the order of the cardinal points is FHH′F′. Since sFand sHare always negative, the object-side cardinal points are always in front of the lens. H′ can take up different positions. Since sH′= −dr2/R = −dr2/{n (r2− r1) + d(n − 1)}, sH′is greater or less than d,i.e.H′ is either in front of or inside the lens, according as d < or > {r2− n(r2− r1)}/(n − 1). (b) If r2< r1the lens is dispersive so long as d < n(r1− r2)/(n − 1). H is always behind S1and H′ behind S2, since sHand sH′are always positive. The focus F is always behind S1and 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(r1− r2)/(n − 1). If d > n(r1− r2)/(n − 1), 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 r1and r2are 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 object- and 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.
Different 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 the lens as in fig. 10. If the object point be in front of the lens, and if it be realized by rays passing from it, it is calledreal. If, on the other hand, the object be behind the lens, it is calledvirtual; 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 isreal, 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 image cannot be intercepted on a screen behind the lens. Such an image is said to bevirtual. These relations are shown in fig. 11.
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 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 lensis 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, 1775); it is not properly a cardinal point, although it has several interesting properties. In fig. 13, let C1P1and C2P2be two parallel radii of a biconvex lens. Join P1P2and let O1P1and O2P2be incident and emergent rays which have P1P2for the path through the lens. Then if M be the intersection of P1P2with the axis, we have angle C1P1M = angle C2P2M; these two angles are—for a ray travelling in the direction O1P1P2O2—the angles of emergence and of incidence respectively. From the similar triangles C2P2M and C1P1M we have
C1M : C2M = C1P1: C2P2= r1: r2.
(11)
Such rays as P1P2therefore divide the distance C1C2in the ratio of the radii,i.e.at the fixed point M, the optical centre. Calling S1M = s1, S2M = s2, then C1S1= C1M + MS1= C1M − S1M,i.e.since C1S1= r1, C1M = r1+ s1, and similarly C2M = r2+ s2. Also S1S2= S1M + MS2= S1M − S2M,i.e.d = s1− s2. Then by using equation (11) we have s1= r1d/(r − r2) and s2= r2d/(r1− r2), and hence s1/s2= r1/r2. The vertex distances of the optical centre are therefore in the ratio of the radii.
The values of s1and s2show that the optical centre of a biconvex or biconcave lens is in the interior of the lens, that in a plano-convex or plano-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 derived 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,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 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 = ½y2/f approximately, whence
f = ½y2/ (n − 1)d.
(12)
In the case of plate-glass (n − 1) = ½ (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 = ½y2(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 (seeAberration); (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).