Achromatism of lenses.

93. By the proper combination of lenses the dispersion of differently coloured lights may approximately be destroyed; for the dispersion produced by one lens may be approximately counteracted by that produced by a second lens, so that the emergent rays may be without colour.

We shall confine our attention to the approximate theory of lenses, in which the thickness of the lens is neglected and the principal points considered as coinciding in one point called the centre of the lens. For the accurate theory of lenses becomes in this case much complicated by the fact that the principal points of the lenses, from which all distances are usually measured, themselves vary in position according to the refractive index of the particular ray we are considering.

In all cases we shall let μ be the refractive index of the standard ray, and u' the refractive index of any other ray. The focal lengths of the lenses will be supposed to be expressed in terms of the refractive index of the standard ray.

It will be useful to find the change in the focal length of a lens, as the ray changes from the standard ray, to any other. The value of the focal length of a double convex lens, the radii of whose bounding surfaces are r, s, respectively, is given by the equation,

where μ is the refractive index of the substance for the standard ray. Giving a small variation to μ, so that it becomes μ', this equation gives

and therefore, if we denote the dispersive power of the medium by, the variation of the focal length is determined by the equation

94. When an image is formed by a lens or system of lenses which is not achromatic, the light being not homogeneous, it will be affected by dispersion in the lenses in two particulars; first, the different coloured images will be distributed in different positions along the axis of the system, and secondly, the coloured images will have different magnitudes. In certain cases both these defects can be removed, in other cases only one of them can be removed, and to choose which correction shall be made, it will be necessary to consider the use to which the system is to be applied, so as to remove the defect which is of the most consequence.

For the object-glass of a telescope two lenses are used, and are placed close together so as to act as one lens. Then a point and its image always lie on the same line through the centre of the lens, so that if the lenses be corrected so that the differently coloured images all lie in the same plane perpendicular to the axis, they will all have the same magnitude. It will therefore be necessary only to make the first correction, and then the other will be satisfied.

These object-glasses are usually made of a double convex lens of crown-glass outside, combined with a double concave lens of flint-glass, which has a higher dispersive power than crown-glass. It is easy to see in a general way how the correction may be effected. By the convex lens the coloured images will be formed at different

distances along the axis, the violet image being the nearest to the lens, and the red image the most remote from it. The effect of the concave lens on these images will be to throw them farther away from the lens, and the effect on the violet image will be stronger than that on the red image. By a proper adjustment of the lenses, the final violet image may be made to coincide with the final red image, or any two other colours may be united in the final image. If the lenses were of the same kind of glass, in order that the dispersion produced by the one should be neutralized by that produced by the other, the lenses would have to be such that the deviation produced by the two lenses would also destroy each other, and therefore the combination would not produce an image at all. But it has been seen that for different kinds of glass the dispersion is not proportional to the deviation, but that flint-glass has a higher dispersive power than crown-glass, so that it is possible to destroy the dispersion without destroying the deviation.

95. We shall now investigate the condition that a combination of two lenses made of different kinds of glass, placed close together, may be achromatic for two given colours.

We shall suppose that one of the colours is the standard colour, and that the focal lengths of the two lenses are f, f', respectively. There will be two images; the first being the image of the object formed by the first lens, and the second being the image of this first image formed by the second lens. Let x, x be the distances of the object and the first image in front of, and behind, the centre of the first lens, y', y the distances of the first and second images in front of, and behind, the centre of the second lens, respectively. Then

If we neglect the thicknesses and the distance between the lenses, y=x, and therefore

The condition that the system should be achromatic is that y should be the same for the two colours; and therefore, since x is independent of the colour,

This is the condition of achromatism for the combination.

This condition is independent of x and y, so that the combination will be achromatic for objects at all distances. It is immaterial in what order the lenses are placed.

In the construction of microscopic object-glasses, achromatic couples of this kind are very generally used, each consisting of a plano-concave lens of flint cemented to a double convex of crown, the plane face being exposed to the incident light.

96. If three thin lenses, formed of media of different dispersive powers, be combined into a single lens, the system may be made achromatic to a higher degree of approximation; the coloured images formed by three different kinds of light may be united. More generally, if n lenses form a combination, whose thickness may be neglected, the system will unite the images formed by rays whose refractive indices are μ and u', provided that

This may be proved in the same way as before. The equation of condition can be satisfied for n-1 systems of

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values of μ- μ, and therefore the images corresponding to n lines of the spectrum may be united.

97. When the two lenses forming a combination are separated by an interval, it is impossible simultaneously to effect the two corrections for dispersion.

For let x, x be the distances of the object and its first image in front of, and behind, the first lens, y', y the distances of the first and final image in front of, and behind, the second lens, respectively, and let B, B,, B' be the linear magnitudes of the object and its images. Then the following ratios must hold:

If the coloured images corresponding to refractive indices μ, μ' be formed at the same distance and also have the same magnitude, we must have x and y fixed and also the ratio xy': x'y. Hence the ratio y': x' is fixed. But x+y=a, where a denotes the distance between the lenses; so that it is necessary that x', y' should both be the same for the two colours. In other words each lens must be achromatic of itself. This cannot be effected unless each lens of the combination be itself an achromatised couple of lenses in contact.

a

98. It is often necessary, however, to correct system of two lenses separated by an interval, for errors