LENS (from Lat. lens, lentil, on account of the similarity of the form of a lens to that of a lentil seed), in optics, an instrument which refracts the luminous rays proceeding from an object in such a manner as to produce an image of the object. It may be regarded as having four principal functions: (1) to produce an image larger than the object, as in the magnifying glass, microscope, &c.; (2) to produce an image smaller than the object, as in the ordinary photographic camera; (3) to convert rays proceeding from a point or other luminous source into a definite pencil, as in lighthouse lenses, the engraver’s globe, &c.; (4) to collect luminous and heating rays into a smaller area, as in the burning glass. A lens made up of two or more lenses cemented together or very close to each other is termed “composite” or “compound”; several lenses arranged in succession at a distance from each other form a “system of lenses,” and if the axes be collinear a “centred system.” This article is concerned with the general theory of lenses, and more particularly with spherical lenses. For a special part of the theory of lenses see [Aberration]; the instruments in which the lenses occur are treated under their own headings.
The most important type of lens is the spherical lens, which is a piece of transparent material bounded by two spherical surfaces, the boundary at the edge being usually cylindrical or conical. The line joining the centres, C1, C2 (fig. 1), of the bounding surfaces is termed the axis; the points S1, S2, at which the axis intersects the surfaces, are termed the “vertices” of the lens; and the distance between the vertices is termed the “thickness.” If the edge be everywhere equidistant from the vertex, the lens is “centred.”
| Fig. 1. |
Although light is really a wave motion in the aether, it is only necessary, in the investigation of the optical properties of systems of lenses, to trace the rectilinear path of the waves, i.e. the direction of the normal to the wave front, and this can be done by purely geometrical methods. It will be assumed that light, so long as it traverses the same medium, always travels in a straight line; and in following out the geometrical theory it will always be assumed that the light travels from left to right; accordingly all distances measured in this direction are positive, while those measured in the opposite direction are negative.
Theory of Optical Representation.—If a pencil of rays, i.e. the totality of the rays proceeding from a luminous point, falls on a lens or lens system, a section of the pencil, determined by the dimensions of the system, will be transmitted. The emergent rays will have directions differing from those of the incident rays, the alteration, however, being such that the transmitted rays are convergent in the “image-point,” just as the incident rays diverge from the “object-point.” With each incident ray is associated an emergent ray; such pairs are termed “conjugate ray pairs.” Similarly we define an object-point and its image-point as “conjugate points”; all object-points lie in the “object-space,” and all image-points lie in the “image-space.”
| Fig. 2. |
The laws of optical representations were first deduced in their most general form by E. Abbe, who assumed (1) that an optical representation always exists, and (2) that to every point in the object-space there corresponds a point in the image-space, these points being mutually convertible by straight rays; in other words, with each object-point is associated one, and only one, image-point, and if the object-point be placed at the image-point, the conjugate point is the original object-point. Such a transformation is termed a “collineation,” since it transforms points into points and straight lines into straight lines. Prior to Abbe, however, James Clerk Maxwell published, in 1856, a geometrical theory of optical representation, but his methods were unknown to Abbe and to his pupils until O. Eppenstein drew attention to them. Although Maxwell’s theory is not so general as Abbe’s, it is used here since its methods permit a simple and convenient deduction of the laws.
Maxwell assumed that two object-planes perpendicular to the axis are represented sharply and similarly in two image-planes also perpendicular to the axis (by “sharply” is meant that the assumed ideal instrument unites all the rays proceeding from an object-point in one of the two planes in its image-point, the rays being generally transmitted by the system). The symmetry of the axis being premised, it is sufficient to deduce laws for a plane containing the axis. In fig. 2 let O1, O2 be the two points in which the perpendicular object-planes meet the axis; and since the axis corresponds to itself, the two conjugate points O′1, O′2, are at the intersections of the two image-planes with the axis. We denote the four planes by the letters O1, O2, and O′1, O′2. If two points A, C be taken in the plane O1, their images are A′, C′ in the plane O′1, and since the planes are represented similarly, we have O′1A′:O1A = O′1C′1:O1C = β1 (say), in which β1 is easily seen to be the linear magnification of the plane-pair O1, O′1. Similarly, if two points B, D be taken in the plane O2 and their images B′, D′ in the plane O′2, we have O′2B′:O2B = O′2D′:O2D = β2 (say), β2 being the linear magnification of the plane-pair O2, O′2. The joins of A and B and of C and D intersect in a point P, and the joins of the conjugate points similarly determine the point P′.
If P′ is the only possible image-point of the object-point P, then the conjugate of every ray passing through P must pass through P′. To prove this, take a third line through P intersecting the planes O1, O2 in the points E, F, and by means of the magnifications β1, β2 determine the conjugate points E′, F′ in the planes O′1, O′2. Since the planes O1, O2 are parallel, then AC/AE = BD/BF; and since these planes are represented similarly in O′1, O′2, then A′C′/A′E′ = B′D′/B′F′. This proportion is only possible when the straight line E′F′ contains the point P′. Since P was any point whatever, it follows that every point of the object-space is represented in one and only one point in the image-space.