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
| 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 x2 is 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 O3O2 and 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
| f − xf ′ − x′ = −xf ′ = −fx′ = γ. | (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 obtain
| 1 | = β / ( | ff ′ | − | ff ′ | ) or 1 = −β | xX | . |
| x − X | x | X | ff ′ |
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′ = H′O′ = H′F′ + F′O′ = F′O′ − F′H′ = x′ − f ′,
whence
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.
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, F1 is the first principal focus of the first, and F′2 the second principal focus of the second system. A ray parallel to the axis at a distance y passes through the second principal focus F′1 of the first system, intersecting the axis at an angle w ′1. The point F′1 will 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′1 in 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, F2 by 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 F1 of 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 ′1 by 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 ′2 in our system of notation, we have
| f ′ = ytan w ′ = yf ′2Δ tan w ′1 = f ′1·f ′2Δ. | (5) |
By taking a ray proceeding from the image-side we obtain for the first principal focal distance of the combination
In the particular case in which Δ = 0, the two focal planes F′1, F2 coincide, 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 arises as to how far this assumption is justified for spherical lenses. To investigate this it is simplest to trace the path of a ray through one spherical
