THE OPTICS OF PHOTOGRAPHIC LENSES 13 



of the symmetry of Eq. 2, the light may travel along a ray in either direction without 

 changing the refraction conditions 



The Action of a Lens. — The action of a lens is illustrated in Fig. 4. If expanding 

 light waves start out from a point source B and travel toward a lens, presently one 

 point of the wave will meet the lens, say at a. Then from a to c the light will travel 

 slowly, while the light from d, which later reaches the rim of the lens, will continue at 

 its original speed. All the intermediate parts of the wave will travel through some 

 air and some glass, so that by the time the light inside the lens has reached c, the light 

 from d will have reached e, and the emerging wave front will be ec as shown. This 

 wave now proceeds onward, and, if the wave happens to be concave as shown, it will 

 shrink to a "focus" at B'. We can imagine an ideal lens in which the emerging wave 

 will be spherical, centered about a single point B', but in practice, owing to the limita- 

 tions imposed by the use of onl}^ spherical surfaces and by the limited availability of 

 optical glass types, the emerging wave will generally not be spherical and there will 



-B-^#^M#F[Sffl-4*i^ 



Fig. 4. — Refraction of light waves by means of a lens. 



not be a perfect focus at B'. In such a case, we say that the lens has "aberrations," 

 the nature of which will be discussed below. 



We may, if we wish, discuss the action of this lens by the ray method by drawing 

 the orthogonals (perpendicular lines) to the wave fronts as in Fig. 4. These rays are 

 there shown dotted in, and it is seen at once that for our "ideal" lens all the rays 

 emerging from it will cross at B', whereas if the lens has aberrations, some of the rays 

 will miss B' and cause a confused patch of light at B' instead of a sharp focus. 



Lens Calculations. — For reasons connected with the manufacturing processes at 

 present in use, only spherical or plane refracting surfaces are used in photographic 

 lenses. Some attempts are being made to employ aspherical surfaces, but these are 

 still entirely experimental. It is a comparatively simple matter to calculate the 

 path of a light ray through a lens system, if the radii of curvature of the surfaces, 

 the thicknesses of the successive lenses, and the refractive indices of the glasses are 

 all given. The formulas by which these calculations are made assume a particularly 

 simple form for the special case of a "paraxial" ray, which is a ray lying very close 

 to the optical axis^ of the lens. For such a ray, if s, s' are the distances of object and 

 image, {i.e., the crossing points of the ray with the axis) from a single refracting surface 

 of radius r separating two mediums having refractive indices n and n', then it can be 

 proved that 



n' _ n n' — n ,„v 



s s r 



By applying this formula successively to all the surfaces in a lens, the position of the 

 final image of a given object point can be determined. 



The signs in this equation are correct if distances are measured outward from the 

 pole of the surface as origin, and are regarded as positive or negative if to the right 

 or left of the surface, respectively. 



1 The "axis" is defined as the line passing through the centers of curvature of all the lens surfaces. 



