One can readily see this chromatic aberration by covering up most of a common reading glass with his hand and looking through the edge portion at a bright light, which will be spread out into a colored band.
If the lens is concave the violet rays will still be the more bent, but now outwards, as shown in Fig. 48. The incident ray a′ is split up and the violet is bent toward v, proceeding as if coming straight from a virtual focus v′ in front of the lens, and nearer it than the corresponding red focus r′. Evidently if we could combine a convex lens, bending the violet inward too much, with a concave one, bending it outward too much, the two opposite variations might compensate each other so that red and violet would come to the same focus—which is the principle of the achromatic objective.
Fig. 48.—Chromatic Aberration of Concave Lens.
If the refractive powers of the lenses were exactly proportional to their dispersive powers, as Newton erroneously thought, it is evident that the concave lens would pitch all the rays outwards to an amount which would annul both the chromatic variation and the total refraction of the convex lens, leaving the pair without power to bring anything to a focus. Fortunately flint glass as compared with crown glass has nearly double the dispersion between red and violet, and only about 20% greater refractive power for the intermediate yellow ray.
Hence, the prismatic dispersive effect being proportional to the total curvature of the lens, the chromatic aberration of a crown glass lens will be cured by a concave flint lens of about half the total curvature, and, the refractions being about as 5 to 6, of ⅗ the total power.
Since the “power” of any lens is the reciprocal of its focal length, a crown glass convex lens of focal length 3, and a concave flint lens of focal length 5 (negative) will form an approximately achromatic combination. The power of the combination will be the algebraic sum of the powers of the components so that the focal length of the pair will be about 5/2 that of the crown lens with which we started.
To be more precise the condition of achromatism is
Σρδn + Σρ′δn′ = 0