L7 Light

Double refraction (Bartolinus 1669)

We return now to the beginning of our study of optics which led us to refraction of light. Let the wall refracting the light be isotropic. Refraction through this wall expresses itself in the law of conservation of the plane of incidence and the constancy of the ratio of sines . We now replace the plate of glass by one of made out of a crystal, in fact - in order to make the phenomena under consideration especially obvious - by a plate of Iceland spar; if it has not been processed, it is, as a rule, like a cube (Fig. 677). We cut the plate parallel to one of the crystal's natural bounding planes, fit them into the shutter in an otherwise dark room and let from the outside fall a parallel bundle of light B on to the plate. If it were glass, we would see the rays emerge from it and cause a spot of light on the opposite wall; However, two bundles of light come out of the Iceland spar and we see on the opposite wall two spots o and ao. This phenomenon is referred to as the double refraction of light. All crystal systems, except for the cubic ones, manifest this, Iceland spar by far most clearly.

Experiments show: The one ray - o - obeys the law of refraction of Snellius (it passes, perpendicularly incident, unrefracted), in general, the other - ao - does not; it does not lie in the plane, determined by the incident ray and the perpendicular and the ratio (sin of angle of incidence)/(sin of angle of refraction) does not have, in general, for every angle of incidence the same value; for example, in Iceland spar, for every ray o at every angle of incidence, nD = 1.658; for the ray ao, it lies in dependence on the angle of incidence between 1,486 and 1. 658. Hence you say that the first ray has been refracted ordinarily, the other extraordinarily. (You see that, for the extraordinary ray, n is occasionally equal to that for the ordinary ray [1.658] and both rays are then refracted in the same way.) How differently from Snellius' law the two rays behave is shown, for example, by turning the Iceland spar plate in the shutter about the perpendicular of incidence (like a wheel about its axle); this does not change at all the ordinarily refracted ray and its corresponding spot of light on the wall. In contrast, the extraordinarily refracted ray rotates about the ordinary ray (also its plane of refraction) and the corresponding sort of light on the wall circulates about that corresponding to the ordinary ray. If we imagine that the crystal plate has been inscribed like the face of a clock, you would see: For the ordinary ray, the position of the plate is not important - whether it is the position in Fig. 679a or b, but this is not true for the extraordinarily refracted ray. What changes as the plate is turned from the one position into the second position and then, continuing to turn it in the same direction, back into the first position? You are dealing with a crystal plate. If you cut from an isotropic block of glass a plate, the direction of cutting is unimportant for its refraction of light. However, for a crystal - unless it happens to belong to the cubic system - the angle between that of the cut and the optical axis is critical. What is understood by the optical axis of a crystal? In a doubly refracting crystal exists a certain direction (in crystals of some systems even two) along which light is only simply, that is, ordinarily refracted. This direction is that of the optical axis of the crystal. If you rotate the plate from a to b (Fig. 679), the incidence of the light changes with respect to the optical axis of the crystal, whence the spot of light corresponding to the extraordinary rays circulates about the other one.

The refraction of light on optically bi-axial crystals is very complicated, whence we will only deal with mono-axial ones, which include Iceland spar. What is understood by the optical axis of Iceland spar? The crystals of Iceland spar are rhombo-hedra (Fig. 677), that is, they have six faces bounded by rhombi and have the appearance of warped cubes. If you make out of equally long bars a model of a cube, the corners and sides of which have hinges, and deform it by pressing on the corners a and b, you obtain a rhombo-hedron. At a and b only obtuse angles meet (at the other corners two sharp angles and one obtuse angle). The straight line through a and b, also one parallel to it, is called the principal axis - it is simultaneouly the optical axis - of the crystal; a cut through the crystal which contains the principal axis and every cut parallel to it is a principal cut.

You cut from the crystal a plate with parallel faces perpendicularly to the optical axis and fit it into the shutter (Fig. 680). The axis then lies horizontally. The bundle of light is to fall at arbitrary angles on to the plate; let the plane of incidence be horizontal. There appear then two bright spots horizontally side by side on the line of intersection of the plane of incidence with the opposite wall. If we rotate the plate about the perpendicular, this action does not affect either of the bright spots - this is a sign that also the extraordinary ray lies in the plane and therefore obeys the the law of Snellius. But it obeys it only in this respect. If you change the angle of incidence and measure every time its magnitude and that of the angle of refraction, you will find for the one ray always nao=1.658, but not for the other ray; the smaller (larger) is the angle of incidence - it is the angle between the ray and the axis, for the axis is perpendicular to the plate (cf. above), that is, parallel to the perpendicular - the larger (smaller) is nao. For the 90º angle of incidence, nao = 1.486. As the angle of incidence (ray/axis) becomes smaller, nao becomes larger, the extraordinary ray then approaches the ordinary one and at the angle of incidence 0º it coincides with it and nao=1.658.

For the -crystal plate, you have the following geometrical characteristic: Its crystal axis always lies in the plane of incidence, or, in other words, the plane of incidence coincides all the time with a principal cut of the crystal. Whenever this happens - you should remember in the following whenever! - one part of the law of Snellius applies: The one concerning the plane of incidence, but not the second: If the angle ray/axis changes, also n changes - the ratio of the sines.

Refraction by an Iceland spar plate with parallel faces

You now cut from the crystal a plate with parallel faces parallel to the optical axis and use it, without any other change, instead of the plate, to start with, in the position when the crystal axis is horizontal, that is, it lies in the (previously assumed to be horizontal) plane of incidence. The plane of incidence then coincides with a principal cut. Their appear again two spots of light o and ao on the wall, horizontally side by side. However, if you rotate this plate about the perpendicular, one spot wanders - the spot ao further away from the perpendicular and caused by the extraordinary ray - in the sense of rotation of the plate and, if the plate has been turned so far that the axis is vertical, that is, has been turned by 90º, it arrives again horizontally beside the other, but further away from the perpendicular than at the start of its motion. This means that the extraordinary ray, to which the moving spot belongs, is refracted more strongly, when the axis lies in the plane of incidence and less strongly when the plane of incidence is perpendicular to the axis.

If the two spots lie side by side horizontally, then both rays lie in the horizontal plane of incidence. Hence, in this respect (maintenance of plane of incidence), the two positions - plane of incidence to the axis and to the axis - do not differ, however, they do with respect to the ratio of the sines. The experiment tells you: In the first position, the axis lies in the plane of incidence. We know already from the plate (Fig. 566 "whenever"), that then, depending on the magnitude of the angle (between 90º and 0º) between the incident ray and the axis, for the extraordinary ray 1.486 < n < 1.658. In the second position, the axis is to the plane of incidence. The experiment tells us that in this position also for the extraordinarily refracted ray sin angle of incidence/ sin angle of refraction is always 1.486 for every angle between 0º and 90º. Thus, in this position of the plane of incidence with respect to the axis of the crystal, the extraordinary ray obeys completely the law of Snellius. If the angle of incidence is 0º, that is, the incident ray is to the crystal plate, then the angle of refraction for the ordinary as well as the extraordinary ray is the angle of refraction is 0, as demanded by the Snellius law; that is, both rays leave the plate in the direction of the incident ray, that is, perpendicularly to the plate and therefore coincide, that is, only one spot of light appears on the wall. Exactly the same occurs when the incident ray meets the plate perpendicularly - at least subjectively - as we see, that is, it is the same. However, it is not objectively the same, because in the second case the refractive index for the extraordinary ray is equal to that for the ordinary ray, that is, there exist actually only a single refracted ray. - However, in the first case, when it is for the ordinary ray as always 1.658, it is for the extraordinary ray 1.486. Actually, there arise two rays, one of which - the extraordinary one - propagates faster than the other, but the eye cannot distinguish between them, whence they seem to be a single ray.

In summary: The ray, extraordinarily refracted by optically mono-axial crystals, obeys, in general, neither of the laws, which the Snellius law demands for ordinarily refracted rays; however, under special conditions - relating to the position of the plane of incidence with respect to the axis of the crystal - it obeys both or at least one of them: It obeys both (maintenance of the plane of incidence and constancy of the ratio of sines), when the plane of incidence is perpendicular to the axis, one (maintenance of plane of incidence) when the axis lies in the plane of incidence.

Positive and negative mono-axial crystals

Iceland spar refracts the ordinary ray more strongly than the extraordinary ray: no = 1.6585, nao = 1.4864. They same occurs with tourmaline, corundum, sapphire, emerald, which are said to be negatively mono-axial. Other crystals such as rock crystal, zircon and ice refract the extraordinary ray more strongly than the ordinary one; they are said to be positively non-axial. For rock crystals - except calcite which is a crystal most frequently employed in Optics - you have no = 1.5442, nao = 1.5533. (Measurements, first performed by Rudberg 1800-1839 1828 with prisms; the refracting edge is cut parallel to the optical axis and during the measurement is placed perpendicularly to the plane of incidence of the light, because then both rays obey the Snellius law. The refractive index of such a crystal prism for both rays is measured in the same way as that of a glass prism.)

The difference between the two refractive indices is for calcspar (0.1721) much larger than for all other crystals, whence it allows most readily observations of double refraction (Fig. 681). You can see it immediately as you replace the wall in Fig. 678 by the retina of your eye. For example, if you place the crystal - it must be very transparent - on the cross (Fig. 682) and then looks through it as perpendicularly as possible, you will, as a rule, see the cross doubled; if you rotate it in the process about the line of viewing, you see that only one cross stays in place, while the other is displaced, indeed in such manner that the one cross stays put while the other circulates. In effect, your observe here that same thing which you have really employed in Fig. 678 for the description of double refraction.

Characteristic properties of doubly refracted light

If double refraction in all non-isotropic substances were so strong that you could see it as readily as in the case of Iceland spar, one could detect immediately whether a substance is isotropic or anisotropic. However, as a rule its is very weak and demands special tools for its detection. Nevertheless, it is a clear indicator regarding the presence of isotropy, because the light after passing through a doubly refracting substance has special properties, which distinguish it fundamentally from ordinary light: It is polarized. If you send polarized light through a substance, you observe certain optical phenomena, which indicate whether it refracts doubly or not (examination under polarized light). For example, you obtain polarized light when you pass ordinary light through a calcspar crystal. However, since two bundles of light emerge and the simultaneous presence of two disturbs, you remove one of them. The prism of William Nicol 1768-1851 1839 serves this purpose and is one of the most important optical tools (Figs. 683/4).

Nicol's prism

This four-sided prism, comprising two three-sided prisms, only allows the extraordinarily refracted ray ao to pass and deflects the ordinarily refracted ray o by total reflection at the plane of contact bc of the two three-sided prisms in such a manner sidewards that it cannot exit; it is absorbed by black paint covering the prism's side faces. You manufacture the prism of Fig. 684 out of a natural calcspar crystal. You split off from it a piece, which is about three times as long as it is broad, and give it to start with a somewhat different shape. In the natural crystal, the end faces form with the edges at a and b angles of 71º. You reduce them by grinding to 68º and bisect this four-sided prism (along bc) by a cut, which is perpendicular to the main cut (here the plane of the drawing) and simultaneously perpendicular to the ground faces ae and gd; you then polish the cut faces bc, stick at them the two halves of the prism together with Canada balsam and obtain Nicol's prism. Fig. 684 displays the course of the rays in it. For yellow light, the refraction index of calcspar (ordinary ray) is 1.658, that of Canada balsam 1.536. Hence the ordinary ray transits at the balsam layer bc from an optically denser into an optically thinner material. The angle of total reflection is 68º. Hence all rays are totally reflected at the balsam layer, the angle of incidence is larger than 68º. A computation confirms: If the rays enter parallel to the prism's edge (or in a direction which does not deviate too much from that direction), the angle of incidence i at the Canada balsam layer is larger than 68º. The refractive index of the extraordinary rays is smaller than that of the balsam layer, whence they are allowed to pass. If you place Nicol's prism instead of the earlier employed plate into the shutter, so that the edge k is horizontal and perpendicular to the shutter, and let a bundle of light mn meet it parallel to k, then there forms on the wall only one spot which belongs to the extraordinary ray. - We will return to Nicol's prism below when dealing with the polarization phenomena of light.

We cannot deal further with the complicated conditions of double refraction. It becomes a more mathematical than physical problem. It is sufficient to point out that you can find geometrically the two refracted rays on the Fresnel wave surface for every crystal - single or double axial - for every given incident ray of light. - While the discussion of double refraction follows logically to that of simple refraction, it is only encountered again during that of polarization of light. Until then, we will only be confronted with simple refraction.

Refraction of light by spherical surface

Geometrical relations between distances of objects and images from the refracting surfaces

The refracting wall which light encounters during its spreading and which affects its further progress was to be plane. How does a curved wall affect its progress (Fig. 685a)? This question takes us to the task of forming an image like with a camera or projector and of supporting the eye by auxiliary tools like spectacles, a magnifying glass, a microscope and a telescope. In these instruments, spherical surfaces are employed most often and we will confine our study to them.

We will resume the approach, which we adopted when we were dealing with the reflection of light by spherical surfaces. Let L on the line LC through the centre C of the surface be a point source of rays - the object point. We follow the ray which encounters at P the spherical border PS between glass and air. The radius CP is the perpendicular at P, the angle of incidence the angle i. The ray remains after refraction in the plane of the drawing; it is refracted towards the axis by the angle of refraction r and intersects it at L'. If you imagine that the drawing has been rotated about LC as axis, you see that all rays on the conical surface (with axis angle PLS), generated by the rotation, pass on refraction through L', which you call the image point of L. You say that L and L' are conjugate points. Since PS is a spherical surface and n sin i = n'·sin r, where n and n' are the absolute refraction indices of air and glass, we can find the position of L' with respect to S. - In the triangle PCL: LC/l = sin i/sinj, in the triangle PCL': L'C/g = sin r/sinj, whence

(LC/L'C)·(g/l) = n'/n (since sin i/sin r = n'/n).

Now assume that P is so close to S that we can substitute l for LS and g for L'S and obtain

(LC/L'C).(L'S/LS) = n'/n. (1)

If we can consider SP to be small, the opening of the bundle of rays is so small that all its rays are approximately perpendicular to the spherical surface (like in the case of a reflecting surface). Equation (1) applies subject to this restrictive assumption. It holds, when light encounters a concave refracting surface, as well as when it enters air from glass (hitherto glass from air), that is, n corresponds to a more strongly refracting substance and n' to a less strongly refracting medium. This is demonstrated by Figs. 685 a,b,c,d, where L is the source of light, C the centre of the sphere, S the vertex of the refracting spherical surface, LP the incident ray, CP the perpendicular for the incident ray, the arrow towards P the ray, the arrow starting at P the ray after refraction and L' the image point - the intersection of the refracted ray with the axis through L and L'. If the ray is refracted towards the axis (a and d), it itself crosses the axis behind the refracting surface (real), if it is refracted away from the axis (b and c), its backwards extension intersects the axis, and indeed in front of the refracting surface (virtual). In all four cases, n is the refractive index of the substance from which comes the light, n' that of the substance into which it enters. In Figs. 865 a,c, n' > n (path of light from air into glass), in Figs. 865 b,d, n' < n ( path from glass into air). In all four cases, i is the angle of incidence, r the refractive angle, that is, Equation (1) applies. Hence you find the distance L'S of the image point from the vertex from the other quantities in all cases from the same equation. But the image point L' lies either on the right or on the left hand side from the vertex. In order to be able to predict where it lies with respect to the vertex, you must know at the same time whether the distance is to be computed towards the left or the right of the vertex. (Similarly, it is not sufficient to know: A temperature lies at a certain distance from zero; you must know simultaneously whether it lies above or below zero.) We will denote again the contrast between + and -, employ again the vertex S as reference point and compute distances, starting from it, as negative to the left hand side and as positive to the right hand side. If we denote the distances L'S by a', LS by a, CS by r, then, for the case of Fig. 685a, (-a + r)/(+a' - r)·(+a'/-a) = n'/n and similarly for every case of Fig. 685, provided you attach to each distance the appropriate sign (+/-) relative to S (as, for example, is shown by the case of Fig. 685d). An elementary algebraic conversion of the left hand side of the formula yields for each case: (n'/a') - (n/a) = (n' - n)/r.

Application of the formula for refraction by spherical surfaces

For the sake of simplicity, replace the absolute refractive indices by their relative values: If n is the absolute refractive index of air, n' that of glass, then the refractive number from air into glass is N = n'/n. Denoting the radius by r, you have

N/a' - 1/a = (N - 1)/r.

Assume for N the value 3/2. If, as in Fig. 685a, the spherical surface is convex towards the left hand side, that is, the centre is on the right hand side of S, and the radius is 50 cm, then r = + 50. If the object point L is on the left hand side of S at a distance of 200 cm, then a = -200. You have now for the image point of L:

(3/2)/a' - 1/-200 = [(3/2) - 1]/(+50) or 3/2a' + 1/200 = 1/100,

whence a' = +300, that is, L' cm on the right hand side of the vertex. If L is only 50 cm from S, then

3/2a' + 1/50 = 1/100, whence a' =-150,

that is, L' also lies, like L,on the left hand side of S. Since it lies on the same side of the vertex as the object point, it is virtual; not the refracted rays intersect themselves, but their backwards extensions. If the spherical surface is concave to the left, then r = -50 and a = -200, whence

(3/2)/a' - 1/-200 = [(3/2) - 1]/(-50), whence a' = -150,

that is, the image point lies also on the left hand side of S and, as in the last numerical example, is virtual.

Introduction of focal length

It is often convenient to rewrite the formula (n'/a') - (n/a) = (n' - n)/r and introduce focal lengths. If the object point L lies infinitely far away from the refracting surface, that is, a = , then the rays meeting it are parallel to each other and the

axis. The image point at which these rays (parallel before refraction) intersect after their refraction - F' in Fig 686 - on the left hand side is called the rear focal point of the refracting surface. Its distance a' from the refracting surface - denoted by s '- follows from n'/a' - n/a = (n' - n)/r. Since a = , n/a = 0, that is a' = nr/(n' - n) s'.

In contrast, if the image point forms on the axis at an infinite distance from the surface, that is, if a' = , the rays after passage through the refracting surface are parallel to each other and the axis. The object point from which these rays - parallel after refraction - come prior to their refraction - F in Fig. 686 right - is called the front focal point of the refracting surface. Its distance a from the refracting surface - denoted by s - is, since a' = , that is, n'/a' = 0,

a = -n·r/(n' - n) s.

You employ the focal lengths s and s' to rewrite the equation n'/a' - n/a = (n' - n)/r. After multiplication of both sides by r/(n' - n), you obtain s'/a' + s/a = 1.

The distances of the focal points from the refracting surface depend only its radius of curvature r and its refractive index. If you employ again instead of the absolute refractive numbers n and n' their relative values, you find s' = N·r/(N - 1) and
s = -r/(N - 1). For example, letting N = 1.5 and r = 3 cm, so that the refracting surface is convex (Fig. 686), then s'=1.5·3/0.5 = 9 cm, s =-3/0.5 = - 6cm. If the refracting surface is concave, that is, r = -3cm, then s' = 1.5·(-3)/0.5 = -9 cm, that is, the rear focal point lies on the same side of the refracting surface as the infinitely far away object point: It is virtual, the backwards extensions of the refracted rays intersect at the focal point (Fig. 687 left). Moreover, a = +6 cm, that is, in order to become parallel after refraction, the rays must must converge to a point lying behind the refracting surface (Fig. 687 right)

* Hence the rear focal point of the concave refracting surface lies (in agreement with the definition) in the sense of the motion of the light ahead of , the front focal point behind the surface.

Elementary bundles of rays of light

The formula n'/a' - n/a = (n' - n)/r interrelates the positions of the object and image points relative to the refracting plane. For its derivation, we have assumed that the angle between the ray LP and the axis is so small, that the rays arrive almost perpendicularly to the spherical surface. Only then is this formula valid. However, if it is true, it applies to every ray of the cone of rays starting from L, because the rays internal to the cone form yet smaller angles with the axis. - A bundle of rays with so small an opening will be called an elementary bundle. If, as in the present case, its principal ray - the ray LC - which you can conceive to be a centroidal axis, meets the sphere perpendicularly, the entire bundle is almost perpendicular to the sphere (normal incident elementary bundle).

We now understand: All the rays of a normally incident elementary bundle of rays pass after refraction through a common point on the principal ray, that is, the bundle is also after refraction homo-centric; the point L is mapped into the point L'.

Mapping of infinitely small objects

What is true for the line LC through the centre of the sphere, is also true for every other line through C: Every point which lies on such a line at a distance a in front of the refracting surface, is mapped into a point behind the refracting surface at a distance a' from the corresponding vertex. Hence, if all points a1, a2, ··· an are at the same distance a from the refracting surface (Fig. 688), their images a'1, a'2, ··· a'n also have the same distance from the corresponding vertex. Hence, if the object points are located on a spherical surface, which is concentric to the refracting sphere, also their images a'1, a'2, ··· a'n lie on such a sphere. If we consider only points in a plane - those lying in the plane of the drawing - we understand that points, which lie on a circular arc, are again mapped onto a circular arc. However, a piece of a spherical surface which is very small, can be viewed to be a plane, a piece of a very short circular arc a straight line, whence we conclude: The element oo of a plane, perpendicular to the axis LC, is mapped into a perpendicular element o'o' of a plane. We see in the drawing only the two lines one of which is the image of the other; you must imagine the drawing being rotated once about the axis LC, in order to survey the process in space.

Briefly speaking: Under the known condition regarding the opening of the incident bundle of rays and the expansion of the object to be mapped - understanding by object an infinitely small line or area element - every point which lies on the axis (through the sphere's centre) is mapped into a point, which also lies on the axis, and a line or plane perpendicular to the axis into a line or plane perpendicular to the axis.

Refraction through a centred sequence of surfaces

The formula n'/a' - n/a = (n' - n)/r only relates to refraction in one plane. In reality, you deal almost always with two - a lens bounded by two spherical surfaces. What happens to a bundle of rays which after passing through the first spherical surface encounters a second, third, etc. surface? We will employ again the absolute refraction index and denote it for the first substance by n, for the second by n', for third by n'', etc. (Fig. 689).Later applications (optical instruments) allows us to assume that the centres of the spherical surfaces lie on a straight line NN. Such a refracting system is said to be centred. Moreover, we will assume that we are dealing again with an elementary bundle which starts from a point L1 of the axis. This bundle remains also homo-centred after refraction at the surface 1 and yields the image point L2 on the axis. All the rays intersect each other at L2 and then encounter the surface 2. The image point L2 is for the surface 2 the object point and has with respect to it the same role as L1 with respect to the surface 1. Also the elementary bundle of rays passing through L2 is refracted homo-centrically to a point of the axis - the point L3, etc. If at one surface, say 3, there is generated a virtual image, then the virtual image point L4 has for the surface 4 the role of the object point on the axis. We see that a (homo-centric) bundle of rays which emits from a point of the axis remains homo-centric after arbitrarily many refractions in a central optical system and generates an image point on the axis.

A lens is formed by a light refracting substance bounded by any pair of spherical surfaces. We are here only concerned with glass lenses, used in optical instruments. There are six different shapes for which pairs of a convex and concave surface and a plane bound the substance; the plane is conceived as a spherical surface with infinite radius. The terminology is shown in Fig. 690.

General formula for lenses

Let a point L on the axis (Fig. 689) emit an elementary bundle to the lens.Where lies the image point L', conjugate to L? It also lies on the axis. The general lens formula tells us whether it lies to the right or to the left hand side of the lens. Assume to start with that the lens is so thin that its thickness can be neglected in comparison with other distances, that is, it is infinitely thin. "This is admissible for a preliminary approximate estimate of the action of an optical system" (Siegfrid Czapski 1861-1907). Accordingly, we measure from the vertex of the first bounding surface also the radius, the image and object distances which relate to the second surface. Denote by a the distance of the object point from the first refracting surface, a' the distance of the image, which arises due to refraction, and r1 its radius; then

n'/a'- n/a = (n' - n)/r1. (1)

This image at the distance a' from the vertex S is now the object for the second surface. Since, by assumption, the vertices of these two surfaces coincide, the object point has from the second surface the distance a'. Let b be the distance of the image, which arises through refraction at the second surface, and r2 the radius. The light leaves the substance with the refractive index n' into the substance with the refractive index n", whence n"/b- n'/a' = (n" - n')/r2. Since n" = n - the lens is bordered on both sides by air! - this equation becomes

n/b- n'/a' = (n' - n)/r2. (2)

Addition of equations (1) and (2) yields

n/b- n/a = (n' - n){1/r1 - 1/r2}, that is, 1/b- 1/a = {(n' - n)/n}{1/r1 - 1/r2}

and with n'/n = N, the general lens formula becomes:

1/b- 1/a = (N - 1){1/r1 - 1/r2}.

This formula becomes simpler by introduction of the focal points of the lens - note: of the lens! The same argument and notation as before for the surface now yield the focal points of the lens. If the source of light is infinitely far away - a = - that is, the rays reach the lens parallel to each other and the axis, its image arises at the rear focal point of the lens. All the rays then pass behind the lens (for diverging lenses extended backwards!) through that point on the axis, the distance f of which (since 1/a=0) is given by 1/f=(N-1){1/r1-1/r2}, whence you can rewrite the general formula in the form:

1/b - 1/a = 1/f.

The front focal point of the lens is the point F on the axis from which the rays must emit (for concave lenses: to which the rays must aim!) so that they only intersect behind the lens at an infinite distance from it (Fig. 691). The planes perpendicular to the axis through the focal points are called the focal planes. The distance of the front (back) focal point of the lens from the first (second) principal plane is called the frontal (rear) focal width. For infinitely thin lenses, the focal width and distance of the focal point are the same.

The general lens formula yields directly the equations for each of the six types of lenses (Fig. 690), you must only be sure to take the signs into account. If the object point lies to the left of the vertex on the axis, a is given a negative sign. If the lens is biconvex, r1 is positive, r2 is negative, if it is biconcave, r1 is negative, r2 is positive, whence for infinitely thin. biconvex/biconcave lenses

1/b + 1/a = (N - 1){1/r1 + 1/r2},

which becomes after introduction of the focal length

1/b + 1/a = f.

Principal planes and principal points (Gauss). Nodal points (Johann Benedict Listing 1808-1882 1845)

You obtain also for lenses of finite thickness a clearly set out formula, if you do not measure the the distances (of object, image, focal points) from the vertex of the refracting surfaces, but from the principal points of Gauss - the two points at which the two principal planes intersect the axis. Fig. 690 shows for each of the six types of lenses the locations of their principal planes and principal points. The corresponding formulae will not be presented here, but the geometrically special position of the principal planes will be discussed, as it assists to follow rays of light through a lens and to find for a given object its image.

Consider a biconvex lens with focal points F and F' (Fig. 692). A ray of light S, which meets parallel to the axis the first surface, passes after refraction at the second surface through F', that is, the ray S and the ray through F' are conjugate. The point of their intersection is at K'. Moreover, imagine there is a second ray S' in the same plane of incidence as S (the plane of the drawing), whicharrives parallel to the axis from the opposite direction at the same height over the axis, that is, in the extension of S. It passes after refraction through F. The two rays intersect at K. Now apply to the ray S' the principle of the invertibility of the paths of rays, that is, imagine it to start at F; this does not alter the geometry. (It has then the direction of the double arrow.) Consider now the points K and K'. K is the point of intersection of S' with the ray through F. However, it is also - and this is now important - the point of intersection of S with the ray through F. Similarly important is that K' is the intersection of S' with the ray through F'. However, we have already seen that the ray S is conjugate to the ray through F' and that S' is conjugate to the ray through F. Hence K' is the point of intersection of two rays which are conjugate to those coming from K. This characterizes K' and K as the conjugate points, that is, when K is an object point, then K' is its image point and inversely. The image and object points lie here on a line parallel to the axis. We have selected arbitrarily the distance from the axis, at which the rays S and S' run parallel to the axis, and have considered only the case of the plane of incidence, which is identical with the plane of the drawing. However, what applies to the points K and K' is also valid for each point of the two lines, which you draw from them perpendicularly to the axis. Moreover, what applies to the plane of incidence - identical with the plane of the drawing - applies to every plane of incidence AA' through the optical axis. If you rotate the drawing once completely about the axis, the lines through K and K' describe two planes in which pairs of points correspond like K and K'. This means that every point of the one plane has its image opposite to it in the other plane, and indeed at the same distance from the axis on the line through it, which is parallel to the axis. - These planes are called principal planes, and the points H and H', at which they are intersected by the axis, principal points. It can be shown that every optical system has only two principal planes.

The focal points and the principal points are important for the geometry and computation of an optical system, whence they are called cardinal points. Given the focal points of the lens and the position of an object, Listing's nodal points also assist with the discovery of the geometrical orientation of an image. N and N' are the nodal points in the biconvex lens (Fig. 693). They are characterized by the following geometrical properties: If a ray is aimed before refraction at the point N on the first refracting surface, it continues after refraction at the second surface of the lens parallel to the direction of incidence, and, indeed, as if it was coming from the second nodal point. - The nodal and principal points coincide, when the first penetrated substance and the last such substance are the same, what is usually the case for the lenses in optical instruments, which are on both sides surrounded by air. For this reason, we will not give details of the nodal points. The eye is an exception, since for it air is the first, but not the last substance.

Images constructed geometrically with the aid of the cardinal points

The following figures demonstrate the significance of the principal planes and nodal points combined with the focal points as geometrical auxiliary tools. Fig. 694a shows a convex lens, E and E' its principal planes, K and K' simultaneously their node points, B and B' their focal points and OP an object. In order to construct the image of OP - to understand geometrically where it lies in the case that it is formed - for example, to find the image point belonging to the object point O, you can employ three of the rays coming from O (two are sufficient!), the paths of which you can construct geometrically from the object to the image with the aid of the six cardinal points. The three rays are:

 ray before refraction property 1. 1 " parallel to axis 2 2 " through front focal point 3. 3 " towards the first nodal point

About the ray 1, we know, firstly, that it passes behind the lens through the focal point B', secondly, before refraction, that it passes through the point h of the first principal plane, whence after refraction it passes through the conjugate point h' of h in the second principal plane, whence its path behind the lens is the line I'I' on which lies the image of O. About the ray 2, we know, firstly, that it passes behind the lens parallel to the axis, secondly, before refraction, that it passes through the point i of the first principal plane, whence it goes after refraction through the point i' of the second principal plane, conjugate to i. Its path behind the lens is therefore the line 2'2'. Hence the image of O also lies on this line. - that is at the intersection of 1'1' and 2'2'. Regarding the ray 3, we know - recalling the property of nodal points! - that it travels behind the lens parallel to the direction which it had ahead of it, indeed through the second nodal point K', whence its path after the lens is the line 3'3'. The image of O lies also on this line, whence it can be determined as the point of intersection of 1'1' and 2'2' or of 1'1' and 3'3' or of 2'2' and 3'3'.

The construction becomes simplest when the lens can be considered to be infinitely thin. The thinner it becomes, the closer approach the principal points each other and the optical centre C. If it is infinitely thin, these points coincide and the main principal planes collapse into the symmetry plane MM' through the optical centre C, perpendicular to the optical axis. For such a lens (Fig. 694b), if you want to construct an image point for a given object point, you must only keep in mind (the notation is the same as in Fig. 694a):

The ray I' passes behind the back focal point and indeed from the same point of the plane MM, at which the ray I has intercepted it - the same, because the two principal planes if Fig. 694a coincide in Fig. 694b. The ray 2' proceeds after the lens parallel to the axis and indeed from the same point of the plane MM at which the ray 2 has intersected it - for the same reason. The ray 3' is the extension of the ray 3, that is, 3 continues through the optical centre C, because C joins both nodal points. The formerly parallel displaced rays 3 and 3' become now one continuous ray through the single remaining nodal point.

Figs. 695 display the construction of the image which disregards the thickness of the lens, a and b for the biconvex, c for the biconcave lens, and illustrate the formulae for the two lenses.

For the biconvex lens (Fig. 695a), you find: If the object lies in front infinitely far away from it (a = ), its image is located at the posterior focal point F' of the lens, is real, inverted and smaller than the object. As the object approaches the lens, its image distances itself from it and grows in size. If the object has approached the lens so that its is only twice the focal distance from it (a = 2f), then also b = 2f and the image is also two focal distances from the lens and as large as the object (Fig. 695a: Object 5 and image 5'). When the object reaches the frontal focal point F, its image is infinitely far away from the lens. If the object comes closer to the lens beyond the focal point F (Fig. 695b), no real image whatsoever arises, but a virtual one which is upright, enlarged and lies on the side of the object. The biconvex lens then enlarges (magnifying glass).

You observe for the biconvex lens (Fig. 695c): If the object is infinitely far away from the lens (a = ), the image is at the back focal point of the lens - on the side of the object - virtual, upright and smaller than the object; as it approaches the lens, also the image comes closer to the lens and grows in size, but remains virtual, upright and smaller than the object. For the biconvex lens, the front focal point lies behind the lens, the rear focal point in front of the lens. (Fig. 687).

However, these are only drawings for the construction, which tell you where and how the images lie and their relationship to their objects, if an image can be realized, however, they do not give information regarding their realization. In fact, we have all along only spoken of the mapping of infinitely small objects and of infinitely narrow bundles of rays, facts which have not been taken into consideration. We will return to this aspect below.

Biconvex lens. Biconcave lens

Everyone knows these lenses; the biconvex lens as burning glass, magnifying glass, also as the large lens of an theatre binocular, the biconcave lens as the small lens of the ordinary theatre binocular. Fig. 695a indicates that the biconvex lens causes the rays, emanating from objects within reach to generate a real image, beyond the lens - on the side of the image (in contrast to on the side of the object) unless, like in Fig. 695b, the object lies closer to the lens than the focal point; they converge to real points of intersection. In contrast, the biconcave lens causes the backwards rays, that is, the prolongations towards the object of the rays which originate from objects within reach, to intersect. Also the refracted rays appear therefore to come from a point on the side of the object; they diverge from virtual points of intersection. Hence you can intercept with a biconvex lens the image on a screen (ground glass plate) and display it like an image; in contrast, through a biconvex lens, if you place your eye in the direction of the rays, you can see the image (as in Fig. 695c), but you cannot intercept it on a screen.

In order to see how differently the two lenses behave towards incident rays, let the sun's rays fall on a sheet of paper and place in their path before they reach the paper one time a biconvex lens, another time a biconcave lens. If you place the former one at a suitable distance from the paper, you will see: 1. The lens, although transparent, creates a circular, strong shadow and 2. its centre (strictly speaking a very small circular disk about its centre) is intensely bright. Strong heat develops there and the paper ignites (focal point). The lens creates a shadow because all rays of light meeting it are directed towards the focal point, whence the points behind the lens do not receive anything. The bright small circular disk is an image of the sun. - If you place the biconcave lens between the sun and the paper, the brightness of the paper is somewhat reduced, because the lens scatters the rays over a larger surface. It is impossible to create an image of the sun! However, if you place the lens in front of your eye, you see the virtual image of the sun on the same side of the lens as the sun (on the side of the object).

Summarizing, we may say: Biconvex lenses refract parallel rays towards each other, biconcave lenses disperse them - the first collect, the second disperse rays. This explains the principal difference in the generation of images. If we look through one or the other at a very far away object, of which you can discern details, you see everything inverted through the biconvex lens - up and down, right and left interchanged. You see someone on a horse, who rides from the right hand side towards the left hand side, move from the left hand side towards the right hand side and the horse's feet up on top, the rider's head at the bottom, and all of it reduced in size. You also see through the biconcave lens the size of everything reduced, but everything in its original relative position. Also this difference originates from the fact that the rays refracted by a biconvex lens intersect behind the lens; the real intersection of the rays causes the inversion, shown by the drawing of the construction.

Achromazy

The focal number N in the lens formula changes with the colour of the light, whence the focal width of a biconvex lens f = r/2(N - 1) has a different length for red from that for yellow or violet. With

 Nred = 1.527 fred = 0.949·r Nviolet = 1.527 fviolet= 0.922·r

you find that (Fig. 696): If you place a screen into the focal point v of the violet rays, the image becomes less sharp by a faded external red coloured band. It arises by the screen also being intercepted by the red rays which aim at their focal point at r as well as from those rays the focal points of which lie between r and v. In contrast, if you move the screen to r, the image losses sharpness by a faded external violet band. It arises from the rays with different colour which have already passed their focal points between r and v. The rays, the focal points of which lie between r and v, generate coloured dispersion circles on the screen - as each individual cone of rays is intercepted by the plane of the screen in a circle of different diameter - at v the red circles are largest, at r the violet circles. The concentric superpositions of the different dispersion circles generate the coloured edges. Hence the lenses of optical instruments (cameras, microscopes, etc.) would yield in non-homogeneous light images with coloured edges unless it were not possible to remove these errors due to colour dispersion. This removal is called achromatization (decolourization), the result of this action achromacy (removal of colours).

Complete achromacy demands union of the foci of the rays of all colours at a single point, which cannot be done technically, and is also unnecessary - the brightness of the different sections of the spectrum differs so much that it becomes sufficient to make the foci of the brightest rays as close together as possible. In order to understand how approximate achromacy is attained, we return to the colour dispersion of white light by a prism.

Achromatic prisms

A prism (Fig. 651) creates a number of coloured images of the opening o - the red one least, the violet one most deflected from the position a, at which the white light would have met the wall opposite the opening. In order to let the violet image coincide with the red one, you need only connect with the prism 2 another one, which is like the first and is placed like 1 in Fig. 697. Every individual coloured image is then displaced by the prism 1 equally far in the direction of the arrow 1 as it was deflected by the prism 2 in the direction of the arrow 2. Then all images appear at the same spot which is only displaced with respect to a by as much as corresponds to dispersion by parallel sided plate - this is what the prisms now form. All exiting rays from the red to the violet ones have thus become parallel to the incident white ray. The chromacy of the image and with it also the change of its form have disappeared - but also the dispersion. We cannot do anything with decolourization, because, if side by side with the colour dispersion also refraction has been cancelled, it is apart from the parallel displacement exactly like it would be without a prism.

But this experiment suggests the direction in which the solution of the task can be found: You must simply produce a prism, which disperses the colours equally strongly as the first prism, which however refracts less and therefore when linked to the first reduces its dispersion, but does not cancel it. For this purpose, you make the first prism out of crown glass, the second out of flint glass*. This difference of the kinds of glass (Fig. 698a) results, for example, in a flint prism with a refracting angle of only 37º in an equally long spectrum, that is, it separates the colours equally strongly as, all else being equal, a crown prism of 60º. In this process, the flint prism deflects the green-yellow rays minimally only by 25º48', the crown prism by 40º. When the two prisms are linked, a deflection of 14º12' remains.

* Crown glass has no lead in it and is in the first place of glass for sheets. Flint glass has much lead; it was initially produced from pulverized flint.

Fig. 698b shows the action of such an achromatic system. White light enters through a small opening in the direction of the arrow. If it were not refracted, it would yield the spot of light W. If you place the prism C on its path (Fig. 699) - the base downwards - it generates the spectrum rV, violet below, red on top. We now introduce with C a prism F (Fig. 700), the

refracting and colour dispersing action of which we define by: If white light arrives from the direction r, in which red light leaves the prism C of Fig. 700 - unrefracted it would create the white spot W' at the location, where the red ray in Fig. 700 meets the wall - it should creat the spectrum r'V' (violet above, red below) and indeed with the same length as C, so that r'V'=rV. If there were not a prism F, the white light in the direction would give a white spot at W', that is, a red spot, a violet one and all the others would have converged at W'. Hence, the prism F has lifted each of these spots: The red one to r' and the violet one by r'V' higher up, which, however, equals rV, and the other colours by corresponding distances. - We place the prism, thus defined, into the position in Fig. 698b between C and the wall. Escaping through C, the bundle rV (Fig. 699) of rays (generated by refraction of the white light) meets F. This bundle would have generated without F the spectrum rV starting at W' on the wall. The spots, which F has to raise, lie therefore from r (Fig. 699), respectively W' ( Fig. 700), onwards apart, that is, V below r by the distance Vr, which however is equal to V'r'. However, since V lifts more by V'r' than it lifts r, but V from the start lies lower by V'r' than r, it lifts V to the same point to which it raises r. Hence the stretched coloured image of the opening becomes the white image W'.

If the prism performs equally perfectly for all colours what it does for red and violet, all coloured images of the opening would be at W', whence the spot W' would have no colour and would be perfectly sharp. However, by the linkage of two prisms made out of the normally employed kinds of glass, you can only cause two colours to cover each other. A combination of two given colours takes place by effectively condensing the spectrum so that they both coincide exactly. This comparison displays immediately the pairwise allocation of short and long wave rays. All rays of different lengths have for the achromatization selected another width of compaction. The phenomenon when the rays of other wave lengths do not have the same compaction width as the two selected ones is called the secondary spectrum. It arises, because in the red section of the spectrum the dispersion of crown glass predominates, in the blue part that of flint glass. Naturally, the colours, which have been selected for the superposition, influence the more or less narrow union of the other colours. The choice of the first depends on the purpose of the optical instrument. For the physiological action of light on the retina, other kinds of rays are decisive than the actinic actions which are important for photography. The optically most active rays lie between the lines C and F; they have maximum brightness between D and E. If you cover C and F by each other, you join simultaneously the brightest parts of the spectrum between D and E. As a result of this good combination of the brightest part, you call the achromasy of C and F or similarly located lines optical and employ it in instruments meant for subjective use. For photographic purposes, the violet actinicly active part of the spectrum is decisive with the maximum of action in the vicinity of G'. In order to place there the minimum of the secondary spectrum, you must join the lines F and ViolHg, when you obtain a pure actinic colour correction. (For astrophysical photography, its comparatively weakly bright objects demand as perfect as possible concentration of the actinically active rays.)

Achromatic lenses

Just as you combine a crown glass prism with a flint glass prism, you combine a crown glass lens with a flint glass lens (Fig. 701). As a rule, they are glued together and the free surface of the concave lens is given a (computable) curvature corresponding to the intended action. Just as the combination of two prisms enables only covering up of two colours, this is also true for the combination of two lenses. In order to join three colours, you must, in general, join at least three lenses. Only a few kinds of Jena glass allow joining of three colours by two lenses, so that a negligible remainder of colour is present.

The achromatization of lenses is decisive for the perfection of microscopes, telescopes and photographic objectives (first, Chester Moor Hall 1703-1771 1733, then John Dolled 1706-1761). The path to glass production came from Fraunhofer who was co-operating since 1809 with Guinand 1748-1824; entirely new and until then completely inaccessible methods were opened up by Ernst Karl Abbe 1840-1905 and Friedrich Otto Schott 1851-1935 1885 by introduction of new components into glass. Initially, only silicic acid had been employed as the basic component of glass flows. Schott introduced also phosphoric and boric acid, phosphate glass types as replacement of crown glass types, boric acid glass types instead of flint glasses. Photography - ordinary as well as micro-photography have been improved by the new lenses: Photographically effective rays intersect at the same location behind the lens at which also the physiological effective rays intersect, whence the photographic image arises at the same location at which the eye sees the image by means of which the equipment is adjusted. Only the coincidence of the optical and the actinic images has made it possible to adjust equipment without tests withg photographic objects.

In the sequel, without special mention, it will be assumed that all lenses are achromatic.