The Microscope. Its History, Construction, and Application 15th ed. / Being a familiar introduction to the use of the instrument, and the study of microscopical science

by Jabez Hogg

Lenses.

Forms of Lenses.—A lens is a portion of a refracting medium bounded by two surfaces which are portions of spheres, having a common axis, termed the axis of the lens. Lenses are distinguished by different names, according to the nature of their surfaces.

Fig. 8.—Converging and Diverging Lenses.

Lenses with sharp edges (thicker at the centre) are convergent or positive lenses. Lenses with blunt edges (thinner at the centre) are divergent or negative lenses. The first group comprises:—(1) The bi-convex lens; (2) the plano-convex lens; (3) the convergent meniscus. The second group:—(4) The concave lens; (5) the plano-concave lens; (6) the divergent meniscus ([Fig. 8]).

Principal Focus.—A lens is usually a solid of revolution, and the axis of revolution is termed the principal axis of the lens. When the surfaces are spherical it is the line joining the centre of curvature.

From the great importance of lenses, especially convex lenses, in practical optics, it will be necessary to explain their properties somewhat at length.

Fig. 9.—Principal Focus of a Convex Lens.

Principal Focus of Convex Lens.—When rays which were originally parallel to the principal axis pass through a convex lens ([Fig. 9]), the effect of the two refractions which they undergo, one on entering and the other on leaving the lens, is to make them all converge approximately to one point F, which is called the principal focus. The distance A F of the principal focus from the lens is called the principal focal distance, or more briefly and usually, the focal length of the lens. The radiant point and its image after refraction are known as the conjugate foci. In every lens the right line perpendicular to the two surfaces is the axis of the lens. This is indicated by the line drawn through the several lenses, as seen in the diagram ([Fig. 8]). The point where the axis cuts the surface of the lens is termed the verte.

Parallel rays falling on a double-convex lens are brought to a focus in the centre of its diameter; conversely, rays diverging from that point are rendered parallel. Hence the focus of a double-convex lens will be at just half the distance, or half the length, of the focus of a plano-convex lens having the same curvature on one side. The distance of the focus from the lens will depend as much on the degree of curvature as upon the refracting power (termed the index of refraction) of the glass of which it may be formed. A lens of crown-glass will have a longer focus than a similar one of flint-glass; since the latter has a greater refracting power than the former. For all ordinary practical purposes we may consider the principal focus—as the focus for parallel rays is termed—of a double-convex lens to be at the distance of its radius, that is, in its centre of curvature; and that of a plano-convex lens to be at the distance of twice its radius, that is, at the other end of the diameter of its sphere of curvature. The converse of all this occurs when divergent rays are made to fall on a convex lens. Rays already converging are brought together at a point nearer than the principal focus; whereas rays diverging from a point within the principal focus are rendered still more diverging, though in a diminished degree. Rays diverging from points more distant than the principal focus on either side, are brought to a focus beyond it: if the point of divergence be within the circle of curvature, the focus of convergence will be beyond it; and vice-versâ. The same principles apply equally to a plano-convex lens; allowance being made for the double distance of its principal focus; and also to a lens whose surfaces have different curvatures; the principal focus of such a lens is found by multiplying the radius of one surface by the radius of the other, and dividing this product by half the sum of the radii.

Fig. 10.—Principal Focus of Concave Lens.

In the case of a concave lens ([Fig. 10]), rays incident parallel to the principal axis diverge after passing through; and their directions, if produced backwards, would approximately meet in a point F; this is its principal focus. It is, however, only a virtual focus, inasmuch as the emergent rays do not actually pass through it, whereas the principal focus of a converging lens is real.

Fig. 11.—Principal Centre of Lens.

Optical Centre of a Lens.—Secondary Axes.—Let O and O′ (Fig. 11) be the centres of the two spherical surfaces of a lens. Draw any two parallel radii, O I, O′ E, to meet these surfaces, and let the joining line I E represent a ray passing through the lens. This ray makes equal angles with the normals at I and E, since these latter are parallel by construction; hence the incident and emergent rays S I, E R also make equal angles with the normals, and are therefore parallel. In fact, if tangent planes (indicated by the dotted lines in the figure) are drawn at I and E, the whole course of the ray S I E R will be the same as if it had passed through a plate bounded by these planes.

Let C be the point in which the line I E cuts the principal axis, and let R, R′ denote the radii of the two spherical surfaces. Then from the similarity of the triangles O C I, O′ C E, we have (O C)/(C O′) = R′/R; which shows that the point C divides the line of centres O O′ in a definite ratio depending only on the radii. Every ray whose direction on emergence is parallel to its direction before entering the lens, must pass through the point C in traversing the lens; and conversely, every ray which in its course through the lens traverses the point C, has parallel directions at incidence and emergence. The point C which possesses this remarkable property is called the centre, or optical centre, of the lens.

This diagram may also be taken to prove my former proposition, that the convex lens is practically a form of two prisms combined.

Fig. 12.—Conjugate Foci, one Real, the other Virtual.

Conjugate Foci, one Real, one Virtual.—When two foci are on the same side of the lens, one (the most distant of the two) must be virtual. For example, in [Fig. 12], if S, S′ are a pair of conjugate foci, one of them S being between the principal focus F and the lens, rays sent to the lens at a luminous point at S, will, after emergence, diverge as if from S′; and rays coming from the other side of the lens, if they converge to S′ before incidence, will in reality be made to meet in S. As S moves towards the lens, S′ moves in the same direction more rapidly; and they become coincident at the surface of the lens.

Formation of Real Images.—Let A B ([Fig. 13]) be an object in front of a lens, at a distance less than the principal focal length. It will have a real image on the other side of the lens. To determine the position of the image by construction, draw through any point A of the object a line parallel to the principal axis, meeting the lens in A′. The ray represented by this line will, after refraction, pass through the principal focus, F, and its intersection with the secondary axis, A O, determines the position of a, the focus conjugate to A. We can in like manner determine the position of b, the focus conjugate to B, another point of the object; and the joining line a b will then be the magnified image of the line A B. It is evident that if a b were the object, A B would be the image.

Fig. 13.—Real and Magnified Image.

The figures 12 and 13 represent the cases in which the distance of the object is respectively greater and less than twice the focal length of the lens.

The focal length of a lens is determined by the convexity of its surfaces and the refractive power of the material of which it is composed, being shortened either by an increase of refractive power, or diminution of the radii of curvature of the faces of the lens. The increase or decrease of spherical aberration is determined by the shape or curvature of the lens; it is less in the bi-convex than in other forms. When a lamp or other source of light is placed at the focus of the rays constituting that portion of its light which falls upon the lens, the light is so refracted as to become parallel. Should the source of light be brought nearer to the lens than the focus the refracted rays are still divergent, though not to the same extent; on the other hand, if the source be beyond the focus, the refracted rays are rendered convergent so as to meet at a point which is mathematically related to the distance of the luminous source from the focus. The former arrangement is that with which we are most familiar, since it is the ordinary magnifying glass.