We have now derived all the properties of the lens in terms of its elements, viz. the refractive index, the radii of the surfaces, and the thickness.
Forms of Lenses.—By varying the signs and relative magnitude of the radii, lenses may be divided into two groups according to their action, and into four groups according to their form.
According to their action, lenses are either collecting, convergent 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.