# An Elementary Treatise on Optics/Chapter 13

CHAP. XIII.
UNEQUAL REFRANGIBILITY OF LIGHT.
122. We come now to an entirely new part of the subject. We have hitherto considered light as something, whether substance or affection of substance, of one single kind. The fact is, however, that light, as it comes to us from the Sun, is not homogeneous, but consists of various kinds, mixed together in different proportions, which being refrangible in different degrees by the same medium, may be exhibited very clearly by causing a Sunbeam to pass through a prism and fall on a sheet of paper placed in a proper position or the wall of a room. The parcels of different kinds of rays being unequally refracted are separated, and the result is a lengthened spectrum, such as is shown in Fig. 138, the colour of which is at one end deep red, and passes through various gradations of orange, yellow, green, and blue to a reddish violet. It appears ​thus that the red rays are the least refrangible, and the violet ones the most.
123. Sir Isaac Newton, who may be considered as the Father of the physical part of this Science, distinguishes seven independent primary colours which he calls red, orange, yellow, green, blue, indigo, and violet; and he tells us that the parts of the spectrum occupied by these colours are proportional to the intervals of the diatonic scale in music. He found their degrees of refrangibility in passing out of glass into air to be as the numbers 77, 771/8, 771/5, 771/3, 771/2, 772/3, 797/8 and 78; those being the values of the sines of refraction to the common sine of incidence 50. Some substances, however, separate the different coloured rays more widely than others, and the dispersive power of media does not appear to depend at all upon their mean refracting power.
​124. The Sun-light once broken by a prism does not admit of any farther decomposition of the same kind; for if a portion of the coloured light of any sort be made to pass through a second prism it preserves its peculiar colour unchanged, and the beam of light, has, after refraction, the same form as before. For this reason the rays of any one colour are called homogeneous, and the light simple or homogeneous light.
125. It is remarkable that an object illuminated by homogeneous light has no colour but that of the particular sort of light falling on it; the only distinction observable being that an object naturally of the same colour as the light, appears brighter than one of a different hue though much less so than a white one. This leads one to conjecture that the colours of natural substances are owing to some power residing in them whereby they decompose the ordinary compound light, and reflect only some particular kinds. This conjecture is strengthened by an experiment made by Sir Isaac Newton, in which he found that when a piece of paper coloured partly red and partly blue, and marked with black lines, was illuminated by a candle and a convex-lens of considerable focal distance placed before it, the images of the two parts were not adjacent, but the blue was at a greater distance from the lens than the red, just as if the paper had been illuminated partly with red and partly with blue homogeneous light.
126. It will readily be conceived that the unequal refrangibility of light must give rise to a good deal of confusion in all images produced by refraction. For instance, a pencil of rays diverging from a point on the axis of a convex-lens will be collected, not to one single focus, but to a series of foci lying at different distances from the lens along its axis, and there will consequently be a confusion analogous to that arising from the aberration which we found to be produced by the spherical form of a refracting surface.
​127. In order to examine this additional confusion we will suppose a beam of Sun-light to fall on a lens, formed so as to collect each kind of homogeneous light accurately to one point without aberration.
Let then QR (Fig. 139.) represent a pencil of compound light incident at R. This will be divided by the refraction into several pencils Rv, Ri, Rb, …, if v, i, b, g, y, o, r, be the points to which the violet, indigo, blue, green, yellow, orange, and red rays converge. And if rays are admitted to all points on the surface of the lens, the points v, i, &c. will be the vertices of so many cones of light of the different colours.
As all the rays do not converge to one point, it is important to know at least where they approach most nearly to it, or where they are all collected in the least space, and how great that space is, which is, in technical language, to require the center and diameter of the least circle of chromatic aberration or dispersion.
A little consideration will easily make it clear that if Bv, bv, Br, br, (Fig. 140.) be the extreme violet and red rays from opposite points of the lens, all the refracted light from the section BAb of the lens will be found in the spaces between the lines Bv, Br, bv, br, all produced without limit, and that the smallest space occupied by them all is the line nmo which joins the intersections of Br, bv; Bv, br respectively: no is therefore the diameter and m the center of the required circle of aberration.
Now mn=AB·mr/Ar; and again, mn=ABmv/Av; so that if we add these together, we shall have
no=Bbvr/Ar+Av=BbAr−Av/Ar+Av,
Am=Av+vm=Av+Av·nm/AB=Av(1+no/Bb)=2Ar·Av/Ar+Av.
$1+r$
$1+v$
${\frac {1}{F}}={\frac {m-1}{\rho }}$
$Ar={\frac {\rho }{r}};\quad Av={\frac {\rho }{v}},$
${\frac {Ar-Av}{Ar+Av}}={\frac {v-r}{v+r}}$
and therefore if we put $a$ for the aperture $Bb,$
$no={\frac {v-r}{v+r}}\cdot a.$
Suppose, for instance, the lens be of crown-glass,
$v=.56$
$r=.54$
${\frac {v-r}{v+r}}={\frac {.02}{1.1}}={\frac {1}{55}}.$
The diameter of the least circle of aberration is therefore ${\frac {1}{55}}$ of the aperture.
128. With regard to the distribution of the light over the surface of the circle of least aberration, it will be sufficient to observe that the vertices of all the cones of coloured light being on the axis of the lens, the center of the circle is one of them, so that it must be strongly illuminated, having the whole of the light of one sort thrown on it, besides portions of the others; the circumference on the contrary is enlightened only by the extreme rays of the red and blue cones, so that it is the least bright part, and it will be easily seen that the light diminishes gradually from the center of the circle to the edge.
On this account the effect of this aberration on images produced by lenses, is not so great as one might imagine from the great magnitude of the least circle of aberration: it certainly substitutes for ​single foci, so many interconfused circles, but as these are bright only at their centers, and above all, as the yellow light, which is the brightest in the spectrum, converges nearly to those centers, the haziness is not very considerable except in cases where the light is very much condensed by a lens of short focal length.
129.The chromatic aberration is a much more serious bar to the perfection of optical instruments depending on the lens, than that owing to the spherical figure, for this latter imperfection can be made quite insensible in most cases, by diminishing the aperture of the lens, since it varies as the square of this line, whereas the former varying as the simple power of the aperture, will be diminished certainly, but very considerably less than the other.
It has therefore been a great desideratum to find some way of constructing a lens, so as to be achromatic, and this has been tolerably well effected, by joining together two or more lenses, made of substances having different dispersive powers, so that the dispersions may be equal and opposite, though the refraction be not wholly destroyed.
130.The expression for the principal focal length of a combination of lenses placed close together was found (p. 68.) to be
${\frac {1}{\phi }}={\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+{\frac {m''-1}{\rho ''}}+....$
If therefore $1+r$ and $1+v$ represent the values of $m$ for red and violet rays, we shall have, taking only two lenses,
${\frac {1}{\phi }}={\frac {r}{\rho }}+{\frac {r'}{\rho '}},$ for the red ray,
${\frac {1}{\phi }}={\frac {v}{\rho }}+{\frac {v'}{\rho '}},$ for the violet.
Now it is clear that if we chuse to leave $\rho$ and $\rho '$ indeterminate, we may equate these two values of $\phi ,$ and so obtain proper values for $\rho$ and $\rho ',$
${\frac {r}{\rho }}+{\frac {r'}{\rho '}}={\frac {v}{\rho }}+{\frac {v'}{\rho '}},$
$r\rho '+r'\rho =v\rho '+v'\rho ,\mathrm {or} \ {\frac {\rho '}{\rho }}=-{\frac {v'-r'}{v-r}},$
which shows that $\rho '$ and $\rho$ must be of different signs, or one lens concave and the other convex; and that they are as the respective dispersions of the lenses.
In order therefore to bring the most unequally refrangible rays, namely, the red and violet, to one focus, we have only to put together a convex and a concave-lens, and to make the quantities represented by $\rho$ and $\rho '$ (or the principal focal lengths, for any one kind of simple light, which are in the same ratio) proportional to the dispersive powers of the substances, of which the lenses are made.
131. The common practice of opticians is to use flint-glass, and crown-glass, the dispersive powers of which are in the ratio of 50 to 33; and therefore a compound-lens such as that represented in Fig. 141. in which the separate focal lengths, for the same kind of homogeneous light, are as 50: 33, will make the red and violet rays of the solar, or any similar light, converge accurately to one point.
To illustrate this, let $v,g,r,$ (Fig. 142.) be the points to which the convex-lens by itself would throw the violet, green, and red rays. The addition of a concave-lens diminishes the convergency, and therefore throws the foci farther off; but it affects the violet and green light more than the red, so that they are all brought closer together, and if the lenses be so matched that the dispersive power of the first is just balanced in all parts by the counter-dispersive power of the second, the rays will all be brought to one single point $f.$
If, however, the substances of which the two lenses are made, do not act with equal inequality, on the different coloured rays, the object will not be attained. If for instance, (Fig. 143.) the first lens disperses the rays so that the foci $v,g,r$ are equidistant, but the second lens acts very nearly as strongly on the green rays as on the violet, it may throw the red focus from $r$ to $r',$ the violet from $v$ to $v'$ close to it, $vv'$ being greater than $rr',$ but the green will go from $g$ to $g'$ making $gg'$ nearly equal to $vv',$ so that the three foci will not coincide.
​Now this is in some degree the case with respect to flint- and crown-glass: they do not disperse the different coloured rays proportionally, and in consequence, if two lenses be matched so that their dispersions are equal and opposite for the extreme rays, there will still be some aberration of green and blue rays uncorrected.
132.Dr. Brewster in his excellent "Treatise on new Philosophical Instruments" details some experiments tending to shew that prisms and lenses of the same substance might be combined so as to correct each other's dispersion, without destroying all the refraction.
He found that when a beam of light passed through a flint-glass prism, so that the deviation was a minimum, (the angles of incidence and emergence being equal,) and the dispersion was corrected by a smaller prism of the same substance, inclined to increase its refraction, the colourless pencil was still considerably refracted from its original direction, by the prism with the greater refracting angle.
This combination, represented in Fig. 144, he proposes to imitate with a pair of lenses, by making them of the form shewn in Fig 145.
The reason why the preceding theory did not lead to any such conclusions as these, appears to be as follows:
It was taken for granted, that a given substance has always the same dispersive power into whatever form it be put, or however its surface be inclined to the light, that is, that the dispersion bears a constant ratio to the mean refraction. Thence it was argued that the dispersion of a lens was as the dispersive power of the substance, and the power of the lens, jointly, or as the dispersive power directly, and focal length inversely, and that therefore the dispersions of a convex and concave-lens might be made equal and opposite, if the fraction dispersive power/focal length was the same in each: but it appears from Dr. Brewster's experiments, that our premises are not true, for that when the angle of incidence is changed, the ratio of refraction is not constant for each kind of primitive light.
133.Dollond, who first constructed achromatic compound-lenses, made them of three different parts, as represented in Fig. 146, ​two convex lenses of crown-glass, with a concave one of white flint-glass between them. In this case we may consider the two outer lenses as producing one single refraction, and the inner one as correcting it.
* ↑ Dr. Wollaston has determined the division of the spectrum, with great accuracy, by looking through a prism at a narrow line of light; his result is that there are four primary colours: red, green, blue, and violet, occupying, respectively, 16, 23, 36 and 25 parts, in length, of the spectrum. The blue light at the bottom of the flame of a candle, is easily resolved into five distinct parcels of the following colours, red, light green, dark green, blue, and violet; that of a spirit lamp, which appears quite blue, consists chiefly of green and violet rays.
* ↑ The difference of the extreme values of the fraction sin inc./sin refr. for glass, appears, according to Newton, to be 1/50. This might be taken as a measure of the dispersive power of the substance, but it is usual to adopt a different expression. When the angles of incidence and refraction are very small, they are nearly proportional to their sines. If therefore we take a constant small angle $\theta$ for the angle of refraction, the angle of incidence will be $m\theta$, and will differ according to the value of $m$. The difference between these two, or ${\overline {m-1\theta }}$, is the refraction; and if $m_{r}$ and $m_{v}$ be values of $m$ for red and violet rays, the difference of the refractions or the dispersion will be ${\overline {m_{v}-1\theta }}-{\overline {m_{r}-1\theta }}$ or $(m_{v}-m_{r})\theta$. Its ratio to the refraction will consequently be ${\frac {m_{v}-m_{r}}{m}}$, taking the mean value for $m$: this is the usual measure of the dispersive power. In flint-glass its value is about 0.05; in crown-glass 0.033.
* ↑ According to this theory we may suppose that a substance appears perfectly white when it reflects all the rays of light indiscriminately, and grey or black when it absorbs some of all, and reflects more or less of the compound light without decomposing or dividing it. Sir Isaac Newton confirmed this by showing that a grey paper placed in sunshine appeared brighter, that is, more white, than a white one placed near it in the shade.
* ↑ We suppose here that there are seven distinct parcels of colours, each refracted to its proper focus, but as in the most perfect experiment of the prismatic spectrum there are no intervals between the colours, the number of foci should probably be infinite.
* ↑ It is quite sufficient to consider the principal focal length, as it will easily be seen, that in the case of rays not parallel, one has only to add ${\frac {1}{\Delta }}$ to the expression.