by purely geometrical methods. It will be assumed that light, so long as it traverses the same medium, always travels in a straight line; and in following out the geometrical theory it will always be assumed that the light travels from left to right; accordingly all distances measured in this direction are positive, while those measured in the opposite direction are negative.
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, O2 be 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 β1 is easily seen to be the linear magnification of the plane-pair O1, O′1. Similarly, if two points B, D be taken in the plane O2 and their images B′, D′ in the plane O′2, we have O′2B′:O2B = O′2D′:O2D = β2 (say), β2 being 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, O2 in the points E, F, and by means of the magnifications β1, β2 determine the conjugate points E′, F′ in the planes O′1, O′2. Since the planes O1, O2 are 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′1 will be determined by the intersection of the conjugate rays C′D′1 and E′F′1, the points D′1, F′1, being readily found from the magnifications β1, β2. Since PP1 is parallel to CE and also to DF, then DF = D1F1. Since the plane O2 is similarly represented in O′2, D′F′ = D′1F′1; this is impossible unless P′P′1 be parallel to C′E′. Therefore every perpendicular object-plane is represented by a perpendicular image-plane.
Let O be the intersection of the line PP1 with the axis, and let O′ be its conjugate; then it may be shown that a fixed magnification β3 exists 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 FF1 and F′F′1 between 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 O1 must 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” (German der bildseitige Brennpunkt, the image-side focus). Similarly if O′4 be on the parallel through O1 but 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” (German der objektseitige Brennpunkt, the object-side focus).
Let H1, H′1 be 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 (O1H1 being conjugate with H′1F′, and O′4H′1 with 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 H1 and 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′1 and 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 H2 must 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′2 parallel 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 O1 to the axis, then OO1 is represented by the line O′O′1 also perpendicular to the axis.
This construction is not applicable if the object or image be infinitely distant. For example, if the object OO1 be 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 passes
