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′ = | y | = | yf′2 | = | f′1·f′2 | . |
| tan w′ | Δ tan w′1 | Δ |
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, 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 refracting 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. |