(11)
Such rays as P1P2 therefore divide the distance C1C2 in the ratio of the radii, i.e. at the fixed point M, the optical centre. Calling S1M = s1, S2M = s2, then C1S1 = C1M + MS1 = C1M − S1M, i.e. since C1S1 = r1, C1M = r1 + s1, and similarly C2M = r2 + s2. Also S1S2 = S1M + MS2 = S1M − S2M, i.e. d = s1 − s2. Then by using equation (11) we have s1 = r1d/(r − r2) and s2 = r2d/(r1 − r2), and hence s1/s2 = r1/r2. The vertex distances of the optical centre are therefore in the ratio of the radii.
The values of s1 and s2 show that the optical centre of a biconvex or biconcave lens is in the interior of the lens, that in a plano-convex or plano-concave lens it is at the vertex of the curved surface, and in a concavo-convex lens outside the lens.
The Wave-theory Derivation of the Focal Length.—The formulae above have been derived by means of geometrical rays. We here give an account of Lord Rayleigh’s wave-theory derivation of the focal length of a convex lens in terms of the aperture, thickness and refractive index (Phil. Mag. 1879 (5) 8, p. 480; 1885, 20, p. 354); the argument is based on the principle that the optical distance from object to image is constant.
| Fig. 14. |
“Taking the case of a convex lens of glass, let us suppose that parallel rays DA, EC, GB (fig. 14) fall upon the lens ACB, and are collected by it to a focus at F. The points D, E, G, equally distant from ACB, lie upon a front of the wave before it impinges upon the lens. The focus is a point at which the different parts of the wave arrive at the same time, and that such a point can exist depends upon the fact that the propagation is slower in glass than in air. The ray ECF is retarded from having to pass through the thickness (d) of glass by the amount (n − 1)d. The ray DAF, which traverses only the extreme edge of the lens, is retarded merely on account of the crookedness of its path, and the amount of the retardation is measured by AF − CF. If F is a focus these retardations must be equal, or AF − CF = (n − 1)d. Now if y be the semi-aperture AC of the lens, and f be the focal length CF, AF − CF = √(f2 + y2) − f = ½y2/f approximately, whence
f = ½y2 / (n − 1)d.
(12)
In the case of plate-glass (n − 1) = ½ (nearly), and then the rule (12) may be thus stated: the semi-aperture is a mean proportional between the focal length and the thickness. The form (12) is in general the more significant, as well as the more practically useful, but we may, of course, express the thickness in terms of the curvatures and semi-aperture by means of d = ½y2 (r1−1 − r2−1). In the preceding statement it has been supposed for simplicity that the lens comes to a sharp edge. If this be not the case we must take as the thickness of the lens the difference of the thicknesses at the centre and at the circumference. In this form the statement is applicable to concave lenses, and we see that the focal length is positive when the lens is thickest at the centre, but negative when the lens is thickest at the edge.”
Regulation of the Rays.