Position of the Cardinal Points of a Lens.—Taking the case of a single spherical refracting surface, and limiting ourselves to the small central aperture, it is seen that the second principal focus F′ is obtained when s is infinitely great. Consequently s′ = -f′; the difference of sign is obvious, since s′ is measured from S, while f′ is measured from F′. The focal lengths are directly deducible from equation (7):—
f′ = −n′r / (n′ − n)
(8)
f = nr / (n′ − n).
(9)
By joining this simple refracting system with a similar one, so that the second spherical surface limits the medium of refractive index n′, we derive the spherical lens. Generally the two spherical surfaces enclose a glass lens, and are bounded on the outside by air of refractive index 1.
The deduction of the cardinal points of a spherical glass lens in air from the relations already proved is readily effected if we regard the lens as a combination of two systems each having one refracting surface, the light passing in the first system from air to glass, and in the second from glass to air. If we know the refractive index of the glass n, the radii r1, r2 of the spherical surfaces, and the distances of the two lens-vertices (or the thickness of the lens d) we can determine all the properties of the lens. A biconvex lens is shown in fig. 8. Let F1 be the first principal focus of the first system of radius r1, and F1′ the second principal focus; and let S1 be its vertex. Denote the distance F1 S1 (the first principal focal length) by f1, and the corresponding distance F′1 S1 by f′1. Let the corresponding quantities in the second system be denoted by the same letters with the suffix 2.
By equations (8) and (9) we have
| f1 = | r1 | , f′1 = − | nr1 | , f2 = − | nr2 | , f′2 = | r2 | , |
| n − 1 | n − 1 | n − 1 | n − 1 |
f2 having the opposite sign to f1. Denoting the distance F′1F2 by Δ, we have Δ = F′1F2 = F′1S1 + S1S2 + S2F2 = F′1S1 + S1S2 − F2S2 = f′1 + d − f2. Substituting for f′1 and f2 we obtain