Now as far as ω4
4 sin2 12ω＝sin2ω + 14sin4ω,
and thus to the same order
QP2＝(u + a sin φ sin ω)2
− a cos φ(u − a cos φ) sin2ω + 14 a(a − u cos φ) sin4 ω.
pose that Q lies on the circle u = a cos φ, the middle term vanishes, and we get, correct as far as ω4,
so that
| QP − u＝a sin φ sin ω + 18a sin φ tan φ sin4 ω | (9), |
in which it is to be noticed that the adjustment necessary to secure the disappearance of sin2ω is sufficient also to destroy the term in sin3ω.
A similar expression can be found for Q′P − Q′A; and thus, if Q′A = v, Q′AO = φ′, where v = a cos φ′, we get
| QP + PQ′ − QA -AQ′＝a sin ω (sin φ − sin φ′) + 18a sin4 ω (sin φ tan φ + sin φ′ tan φ′) |
(10). |
If φ′ = φ, the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q′ are conjugate foci, but also that the foci are exempt from the most important term in the aberration. In the present application φ′ is not necessarily equal to φ; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to ± mλ (m integral), and therefore without influence, provided
| σ (sin φ − sinφ′)＝± mλ | (11), |
where σ denotes the constant interval between the planes containing the lines. This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions. They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ′A, and the outstanding aberration is of the fourth order.
In order that a large part of the field of view may be in focus at once, it is desirable that the locus of the focused spectrum should be nearly perpendicular to the line of vision. For this purpose Rowland places the eye-piece at O, so that φ = 0, and then by (11) the value of φ′ in the mth spectrum is
| σ sin φ′＝± mλ | (12). |
If ω now relate to the edge of the grating, on which there are altogether n lines,
nσ＝2a sin ω,
and the value of the last term in (10) becomes
116nσsin3 ω sin φ′ tan φ′,
or
| 116mnλ sin3ω tan φ′ | (13). |
This expresses the retardation of the extreme relatively to the central ray, and is to be reckoned positive, whatever may be the signs of ω, and φ′. If the semi-angular aperture (ω) be 1100, and tan φ′ = 1, mn might be as great as four millions before the error of phase would reach 14λ. If it were desired to use an angular aperture so large that the aberration according to (13) would be injurious, Rowland points out that on his machine there would be no difficulty in applying a remedy by making σ slightly variable towards the edges. Or, retaining σ constant, we might attain compensation by so polishing the surface as to bring the circumference slightly forward in comparison with the position it would occupy upon a true sphere.
It may be remarked that these calculations apply to the rays in the primary plane only. The image is greatly affected with astigmatism; but this is of little consequence, if γ in (8) be small enough. Curvature of the primary focal line having a very injurious effect upon definition, it may be inferred from the excellent performance of these gratings that γ is in fact small. Its value does not appear to have been calculated. The other coefficients in (8) vanish in virtue of the symmetry.
The mechanical arrangements for maintaining the focus are of great simplicity. The grating at A and the eye-piece at O are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other. A tie between the middle point of the rod OA and Q can be used if thought desirable.
The absence of chromatic aberration gives a great advantage in the comparison of overlapping spectra, which Rowland has turned to excellent account in his determinations of the relative wave-lengths of lines in the solar spectrum (Phil. Mag., 1887).
For absolute determinations of wave-lengths plane gratings are used. It is found (Bell, Phil. Mag., 1887) that the angular measurements present less difficulty than the comparison of the grating interval with the standard metre. There is also some uncertainty as to the actual temperature of the grating when in use. In order to minimize the heating action of the light, it might be submitted to a preliminary prismatic analysis before it reaches the slit of the spectrometer, after the manner of Helmholtz.
In spite of the many improvements introduced by Rowland and of the care with which his observations were made, recent workers have come to the conclusion that errors of unexpected amount have crept into his measurements of wave-lengths, and there is even a disposition to discard the grating altogether for fundamental work in favour of the so-called “interference methods,” as developed by A. A. Michelson, and by C. Fabry and J. B. Pérot. The grating would in any case retain its utility for the reference of new lines to standards otherwise fixed. For such standards a relative accuracy of at least one part in a million seems now to be attainable.
Since the time of Fraunhofer many skilled mechanicians have given their attention to the ruling of gratings. Those of Nobert were employed by A. J. Ångström in his celebrated researches upon wave-lengths. L. M. Rutherfurd introduced into common use the reflection grating, finding that speculum metal was less trying than glass to the diamond point, upon the permanence of which so much depends. In Rowland’s dividing engine the screws were prepared by a special process devised by him, and the resulting gratings, plane and concave, have supplied the means for much of the best modern optical work. It would seem, however, that further improvements are not excluded.
There are various copying processes by which it is possible to reproduce an original ruling in more or less perfection. The earliest is that of Quincke, who coated a glass grating with a chemical silver deposit, subsequently thickened with copper in an electrolytic bath. The metallic plate thus produced formed, when stripped from its support, a reflection grating reproducing many of the characteristics of the original. It is best to commence the electrolytic thickening in a silver acetate bath. At the present time excellent reproductions of Rowland’s speculum gratings are on the market (Thorp, Ives, Wallace), prepared, after a suggestion of Sir David Brewster, by coating the original with a varnish, e.g. of celluloid. Much skill is required to secure that the film when stripped shall remain undeformed.
A much easier method, applicable to glass originals, is that of photographic reproduction by contact printing. In several papers dating from 1872, Lord Rayleigh (see Collected Papers, i. 157, 160, 199, 504; iv. 226) has shown that success may be attained by a variety of processes, including bichromated gelatin and the old bitumen process, and has investigated the effect of imperfect approximation during the exposure between the prepared plate and the original. For many purposes the copies, containing lines up to 10,000 to the inch, are not inferior. It is to be desired that transparent gratings should be obtained from first-class ruling machines. To save the diamond point it might be possible to use something softer than ordinary glass as the material of the plate.
9. Talbot’s Bands.—These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, half the aperture being covered by a thin plate, e.g. of glass or mica. The view of the matter taken by the discoverer (Phil. Mag., 1837, 10, p. 364) was that any ray which suffered in traversing the plate a retardation of an odd number of half wave-lengths would be extinguished, and that thus the spectrum would be seen interrupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago’s theory of stellar scintillation.[1] It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a half-wave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of
- ↑ On account of inequalities in the atmosphere giving a variable refraction, the light from a star would be irregularly distributed over a screen. The experiment is easily made on a laboratory scale, with a small source of light, the rays from which, in their course towards a rather distant screen, are disturbed by the neighbourhood of a heated body. At a moment when the eye, or object-glass of a telescope, occupies a dark position, the star vanishes. A fraction of a second later the aperture occupies a bright place, and the star reappears. According to this view the chromatic effects depend entirely upon atmospheric dispersion.
