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 through 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 the lateral magnification. Denoting this by β, we have