crystals of potassium chlorate. Stokes showed that the reflected light is often in a high degree monochromatic, and that it is connected with the existence of twin planes. A closer discussion appears to show that the twin planes must be repeated in a periodic manner (Phil. Mag., 1888, 26, 241, 256; also see R. W. Wood, Phil. Mag., 1906).
A beautiful example of a similar effect is presented by G. Lippmann’s coloured photographs. In this case the periodic structure is actually the product of the action of light. The plate is exposed to stationary waves, resulting from the incidence of light upon a reflecting surface (see Photography).
All that can be expected from a physical theory is the determination of the composition of the light reflected from or transmitted by a thin plate in terms of the composition of the incident light. The further question of the chromatic character of the mixtures thus obtained belongs rather to physiological optics, and cannot be answered without a complete knowledge of the chromatic relations of the spectral colours themselves. Experiments upon this subject have been made by various observers, and especially by J. Clerk Maxwell (Phil. Trans., 1860), who has exhibited his results on a colour diagram as used by Newton. A calculation of the colours of thin plates, based upon Maxwell’s data, and accompanied by a drawing showing the curve representative of the entire series up to the fifth order, has been given by Rayleigh (Edin. Trans., 1887). The colours of Newton’s scale are met with also in the light transmitted by a somewhat thin plate of doubly-refracting material, such as mica, the plane of analysis being perpendicular to that of primitive polarization.
The same series of colours occur also in other optical experiments, e.g. at the centre of the illuminated area when light issuing from a point passes through a small round aperture in an otherwise opaque screen.
The colours of which we have been speaking are those formed at nearly perpendicular incidence, so that the retardation (reckoned as a distance), viz. 2μt cos α′, as sensibly independent of λ. This state of things may be greatly departed from when the thin plate is rarer than its surroundings, and the incidence is such that α′ is nearly equal to 90°, for then, in consequence of the powerful dispersion, cos α′ may vary greatly as we pass from one colour to another. Under these circumstances the series of colours entirely alters its character, and the bands (corresponding to a graduated thickness) may even lose their coloration, becoming sensibly black and white through many alternations (Newton’s Opticks, bk. ii.; Fox-Talbot, Phil. Mag., 1836, 9, p. 40l). The general explanation of this remarkable phenomenon was suggested by Newton.
Let us suppose that plane waves of white light travelling in glass are incident at angle α upon a plate of air, which is bounded again on the other side by glass. If μ be the index of the glass, α′ the angle of refraction, then sin α′ = μ sin α; and the retardation, expressed by the equivalent distance in air, is
and the retardation in phase is 2t cos α′/λ, λ being as usual the wave-length in air.
The first thing to be noticed is that, when α approaches the critical angle, cosα′ becomes as small as we please, and that consequently the retardation corresponding to a given thickness is very much less than at perpendicular incidence. Hence the glass surfaces need not be so close as usual.
A second feature is the increased brilliancy of the light. According to (7) the intensity of the reflected light when at a maximum (sin 12κγ = 1) is 4e2/(1 + e2)2. At perpendicular incidence e is about 15, and the intensity is somewhat small; but, as cos α′ approaches zero, e approaches unity, and the brilliancy is much increased.
But the peculiarity which most demands attention is the lessened influence of a variation in λ upon the phase-retardation. A diminution of λ of itself increases the retardation of phase, but, since waves of shorter wave-length are more refrangible, this effect may be more or less perfectly compensated by the greater obliquity, and consequent diminution in the value of cos α′. We will investigate the conditions under which the retardation of phase is stationary in spite of a variation of λ.
In order that λ−1 cos α′ may be stationary, we must have
where (α being constant)
Thus
| cot2 α′ = λμ dμdλ | (9), |
giving α′ when the relation between μ and λ is known.
According to A. L. Cauchy’s formula, which represents the facts very well throughout most of the visible spectrum,
| μ = A+Bλ−2 | (10), |
so that
| cot2 α′ = 2Bλ2μ 2(μ − A)μ | (11). |
If we take, as for Chance’s “extra-dense flint,” B = .984 × 10−10, and as for the soda lines, μ = 1.65, λ = 5.89 × 10−6, we get
At this angle of refraction, and with this kind of glass, the retardation of phase is accordingly nearly independent of wave-length, and therefore the bands formed, as the thickness varies, are approximately achromatic. Perfect achromatism would be possible only under a law of dispersion
If the source of light be distant and very small, the black bands are wonderfully fine and numerous. The experiment is best made (after Newton) with a right-angled prism, whose hypothenusal surface may be brought into approximate contact with a plate of black glass. The bands should be observed with a convex lens, of about 8 in. focus. If the eye be at twice this distance from the prism, and the lens be held midway between, the advantages are combined of a large field and of maximum distinctness.
If Newton’s rings are examined through a prism, some very remarkable phenomena are exhibited, described in his twenty-fourth observation (Opticks; see also Place, Pogg. Ann., 1861, 114, 504). “When the two object-glasses are laid upon one another, so as to make the rings of the colours appear, though with my naked eye I could not discern above eight or nine of those rings, yet by viewing them through a prism I could see a far greater multitude, insomuch that I could number more than forty.... And I believe that the experiment may be improved to the discovery of far greater numbers.... But it was on but one side of these rings, namely, that towards which the refraction was made, which by the refraction was rendered distinct, and the other side became more confused than when viewed with the naked eye....
“I have sometimes so laid one object-glass upon the other that to the naked eye they have all over seemed uniformly white, without the least appearance of any of the coloured rings; and yet by viewing them through a prism great multitudes of those rings have discovered themselves.”
Newton was evidently much struck with these “so odd circumstances”; and he explains the occurrence of the rings at unusual thicknesses as due to the dispersing power of the prism. The blue system being more refracted than the red, it is possible under certain conditions that the nth blue ring may be so much displaced relatively to the corresponding red ring as at one part of the circumference to compensate for the different diameters. A white stripe may thus be formed in a situation where without the prism the mixture of colours would be complete, so far as could be judged by the eye.
The simplest case that can be considered is when the “thin plate” is bounded by plane surfaces inclined to one another at a small angle. By drawing back the prism (whose edge is parallel to the intersection of the above-mentioned planes) it will always be possible so to adjust the effective dispersing power as to bring the nth bars to coincidence for any two assigned colours, and therefore approximately for the entire spectrum. The formation of the achromatic band, or rather central black band, depends indeed upon the same principles as the fictitious shifting of the centre of a system of Fresnel’s bands when viewed through a prism.
But neither Newton nor, as would appear, any of his successors has explained why the bands should be more numerous than usual, and under certain conditions sensibly achromatic for a large number of alternations. It is evident that, in the particular case of the wedge-shaped plate above specified, such a result would not occur. The width of the bands for any colour would be proportional to λ, as well after the displacement by the prism as before; and the succession of colours formed in white light and the number of perceptible bands would be much as usual.
The peculiarity to be explained appears to depend upon the curvature of the surfaces bounding the plate. For simplicity suppose that the lower surface is plane (y = 0), and that the approximate equation of the upper surface is y = a + bx2, a being thus the least distance between the plates. The black of the nth order for wave-length λ occurs when
| 12nλ = a+bx2 | (12); |
and thus the width (δx) at this place of the band is given by
| 12λ = 2bxδx | (13); |
or
| δx = λ4bx = λ4√b · √(12nλ − a) | (14). |
If the glasses be in contact, as is usually supposed in the theory of Newton’s rings, a = 0, and δx∞λ12, or the width of the band of the nth order varies as the square root of the wave-length, instead of as the first power. Even in this case the overlapping and subsequent obliteration of the bands is greatly retarded by the use of the prism, but the full development of the phenomenon requires that α should be finite. Let us inquire what is the condition in order that the width of the band of the nth order may be stationary, as λ varies. By (14) it is necessary that the variation of λ2/(12nλ − a) should vanish. Hence a = 14nλ, so that the interval between the surfaces at the place where the nth band is formed should be half due to curvature and half to imperfect contact at the place of closest approach. If this condition be satisfied, the achromatism of the nth band, effected by the prism, carries with it the achromatism of a large number of neighbouring bands, and thus gives rise to the remarkable effects described by Newton. Further developments
