and thus to the same order
QP^2 = (u + a sin [phi] sin [omega])^2
-a cos [phi](u - a cos [phi]) sin^2[omega] + 1/4 a(a - u cos[phi]) sin^4 [omega].
But if we now suppose that Q lies on the circle u = a cos [phi], the
middle term vanishes, and we get, correct as far as [omega]^4,
/ / a^2 sin^2[phi] sin^4[omega]\
QP = (u + a sin[phi] sin[omega]) / ( 1 + --------------------------- );
\/ \ 4u /
so that
QP - u = a sin [phi] sin [omega] + 1/8 a sin[phi] tan[phi] sin^4 [omega] (9),
in which it is to be noticed that the adjustment necessary to secure
the disappearance of sin^2[omega] is sufficient also to destroy the
term in sin^3[omega].
A similar expression can be found for Q'P - Q'A; and thus, if Q'A = v,
Q'AO = [phi]', where v = a cos [phi]', we get
QP + PQ' - QA -AQ' = a sin[omega] (sin[phi] - sin[phi]')
+ 1/8 a sin^4 [omega] (sin[phi] tan[phi] + sin[phi]' tan[phi]') (10).
If [phi]' = [phi], 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 [phi]' is not necessarily equal
to [phi]; 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 [-+]
m[lambda] (m integral), and therefore without influence, provided
[sigma] (sin[phi] - sin[phi]') = [+-] m[lambda] (11),
where [sigma] 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 [phi] = 0, and then by (11) the
value of [phi]' in the m^th spectrum is
[sigma] sin [phi]' = [+-] m[lambda] (12).
If [omega] now relate to the edge of the grating, on which there are
altogether n lines,
n[sigma] = 2a sin [omega],
and the value of the last term in (10) becomes
1/16 n[sigma] sin^3[omega] sin[phi]' tan[phi]',
or
1/16 mn[lambda] sin^3[omega] tan [phi]' (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 [omega], and [phi]'. If the semi-angular aperture ([omega]) be
1/100, and tan [phi]' = 1, mn might be as great as four millions
before the error of phase would reach 1/4[lambda]. 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 [sigma] slightly
variable towards the edges. Or, retaining [sigma] 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 [gamma] 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 [gamma] 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. Perot. 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. Angstrom 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.[9] 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 the bands comes about in
a very curious way, as is shown by a circumstance first observed by
Brewster. When the retarding plate is held on the side towards the red
of the spectrum, _the bands are not seen_. Even in the contrary case,
the thickness of the plate must not exceed a certain limit, dependent
upon the purity of the spectrum. A satisfactory explanation of these
bands was first given by Airy (_Phil. Trans._, 1840, 225; 1841, 1), but
we shall here follow the investigation of Sir G. G. Stokes (_Phil.
Trans._, 1848, 227), limiting ourselves, however, to the case where the
retarded and unretarded beams are contiguous and of equal width.
The aperture of the unretarded beam may thus be taken to be limited by
x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be
given by x = 0, x = h, y= -l, y = +l. For the former (1) S 3 gives
_ _
1 / 0 / +l / x[xi] + y[eta]\
- --------- | | sin k (at - f + -------------- )dxdy
[lambda]f _/-h _/-l \ f /
2lh f k[eta]l 2f k[xi]h / [xi]h \
= - --------- . ------- sin ------- . ------ sin ------ . sin k (at - f - ----- ) (1),
[lambda]f k[eta]l f k[xi]h 2f \ 2f /
on integration and reduction.
For the retarded stream the only difference is that we must subtract R
from at, and that the limits of x are 0 and +h. We thus get for the
disturbance at [xi], [eta], due to this stream
2lh f k[eta]l 2f k[xi]h / [xi]h \
- --------- . ------- sin ------- . ------ sin ------ . sin k (at - f - R + ----- ) (2).
[lambda]f k[eta]l f k[xi]h 2f \ 2f /
If we put for shortness [pi] for the quantity under the last circular
function in (1), the expressions (1), (2) may be put under the forms u
sin [tau], v sin ([tau] - [alpha]) respectively; and, if I be the
intensity, I will be measured by the sum of the squares of the
coefficients of sin [tau] and cos [tau] in the expression
u sin[tau] + v sin([tau] - [alpha]),
so that