and condensing, or divergent and dispersing; the term positive is
sometimes applied to the former, and the term negative to the
latter. Convergent lenses transform a parallel pencil into a converging
one, and increase the convergence, and diminish the divergence
of any pencil. Divergent lenses, on the other hand, transform
a parallel pencil into a diverging one, and diminish the convergence,
and increase the divergence of any pencil. In convergent lenses the
first principal focal distance is positive and the second principal
focal distance negative; in divergent lenses the converse holds.
The four forms of lenses are interpretable by means of equation (10).
| f = | r1r2n | . |
| (n−1) { n (r2−r1) + d(n−1)} |
Fig. 9.
(1) If r1 be positive and r2 negative. This type is called biconvex (fig. 9, 1). The first principal focus is in front of the lens, and the second principal focus behind the lens, and the two principal points are inside the lens. The order of the cardinal points is therefore FS1HH′S2F′. The lens is convergent so long as the thickness is less than n(r1−r2)/(n−1). The special case when one of the radii is infinite, in other words, when one of the bounding surfaces is plane is shown in fig. 9, 2. Such a collective lens is termed plano-convex. As d increases, F and H move to the right and F′ and H′ to the left. If d = n(r1−r2)/(n−1), the focal length is infinite, i.e. the lens is telescopic. If the thickness be greater than n(r1−r2)/(n−1), the lens is dispersive, and the order of the cardinal points is HFS1S2F′H′.
(2) If r1 is negative and r2 positive. This type is called biconcave (fig. 9, 4). Such lenses are dispersive for all thicknesses. If d increases, the radii remaining constant, the focal lengths diminish. It is seen from the equations giving the distances of the cardinal points from the vertices that the first principal focus F is always behind S1, and the second principal focus F′ always in front of S2, and that the principal points are within the lens, H′ always following H. If one of the radii becomes infinite, the lens is plano-concave (fig. 9, 5).
(3) If the radii are both positive. These lenses are called convexo-concave. Two cases occur according as r2 > r1, or < r1. (a) If r2 > r1, we obtain the mensicus (fig. 9, 3). Such lenses are always collective; and the order of the cardinal points is FHH′F′. Since sF and sH are always negative, the object-side cardinal points are always in front of the lens. H′ can take up different positions. Since sH′ = −dr2/R = −dr2/{n (r2−r1) + d(n−1)}, sH′ is greater or less than d, i.e. H′ is either in front of or inside the lens, according as d < or > {r2−n(r2−r1)}/(n−1). (b) If r2 < r1 the lens is dispersive so long as d < n(r1−r2)/(n−1). H is always behind S1 and H′ behind S2, since sH and sH′ are always positive. The focus F is always behind S1 and F′ in front of S2. If the thickness be small, the order of the cardinal points is F′HH′F; a dispersive lens of this type is shown in fig. 9, 6. As the thickness increases, H, H′ and F move to the right, F more rapidly than H, and H more rapidly than H′; F′, on the other hand, moves to the left. As with biconvex lenses, a telescopic lens, having all the cardinal points at infinity, results when d = n(r1−r2)/(n−1). If d > n(r1−r2)/(n−1), f is positive and the lens is collective. The cardinal points are in the same order as in the mensicus, viz. FHH′F′; and the relation of the principal points to the vertices is also the same as in the mensicus.
(4) If r1 and r2 are both negative. This case is reduced to (3) above, by assuming a change in the direction of the light, or, in other words, by interchanging the object- and image-spaces.
The six forms shown in fig. 9 are all used in optical constructions. It may be stated fairly generally that lenses which are thicker at the middle are collective, while those which are thinnest at the middle are dispersive.
Fig. 10.
Different Positions of Object and Image.—The principal points are always near the surfaces limiting the lens, and consequently the lens divides the direct pencil containing the axis into two parts. The object can be either in front of or behind the lens as in fig. 10. If the object point be in front of the lens, and if it be realized by rays passing from it, it is called real. If, on the other hand, the object be behind the lens, it is called virtual; it does not actually exist, and can only be realized as an image.
Fig. 11.
When we speak of “object-points,” it is always understood that the rays from the object traverse the first surface of the lens before meeting the second. In the same way, images may be either real or virtual. If the image be behind the second surface, it is real, and can be intercepted on a screen. If, however, it be in front of the lens, it is visible to an eye placed behind the lens, although the rays do not actually intersect, but only appear to do so, but the image cannot be intercepted on a screen behind the lens. Such an image is said to be virtual. These relations are shown in fig. 11.
Fig. 12.
By referring to the equations given above, it is seen that a thin convergent lens produces both real and virtual images of real objects, but only a real image of a virtual object, whilst a divergent lens produces a virtual image of a real object and both real and virtual images of a virtual object. The construction of a real image of a real object by a convergent lens is shown in fig. 3; and that of a virtual image of a real object by a divergent lens in fig. 12.
Fig. 13.
The optical centre of a lens is a point such that, for any ray which passes through it, the incident and emergent rays are parallel. The idea of the optical centre was originally due to J. Harris (Treatise on Optics, 1775); it is not properly a cardinal point, although it has several interesting properties. In fig. 13, let C1P1 and C2P2 be two parallel radii of a biconvex lens. Join P1P2 and let O1P1 and O2P2 be incident and emergent rays which have P1P2 for the path through the lens. Then if M be the intersection of P1P2 with the axis, we have angle C1P1M = angle C2P2M; these two angles are—for a ray travelling in the direction O1P1P2O2—the angles of emergence and of incidence respectively. From the similar triangles C2P2M and C1P1M we have
| C1M : C2M = C1P1 : C2P2 = r1 : r2. | (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,
