The effect of a gradual increase in the interval (fig. 9) as we pass
across the grating has been investigated by M. A. Cornu (_C.R._, 1875,
80, p. 655), who thus explains an anomaly observed by E. E. N.
Mascart. The latter found that certain gratings exercised a converging
power upon the spectra formed upon one side, and a corresponding
diverging power upon the spectra on the other side. Let us suppose
that the light is incident perpendicularly, and that the grating
interval increases from the centre towards that edge which lies
nearest to the spectrum under observation, and decreases towards the
hinder edge. It is evident that the waves from _both_ halves of the
grating are accelerated in an increasing degree, as we pass from the
centre outwards, as compared with the phase they would possess were
the central value of the grating interval maintained throughout. The
irregularity of spacing has thus the effect of a convex lens, which
accelerates the marginal relatively to the central rays. On the other
side the effect is reversed. This kind of irregularity may clearly be
present in a degree surpassing the usual limits, without loss of
definition, when the telescope is focused so as to secure the best
effect.
It may be worth while to examine further the other variations from
correct ruling which correspond to the various terms expressing the
deviation of the wave-surface from a perfect plane. If x and y be
co-ordinates in the plane of the wave-surface, the axis of y being
parallel to the lines of the grating, and the origin corresponding to
the centre of the beam, we may take as an approximate equation to the
wave-surface
x^2 y^2
z = ------ + Bxy + ------- + [alpha]x^3 + [beta]x^2y + [gamma]xy^2 + [delta]y^3 + ... (8);
2[rho] 2[rho]'
and, as we have just seen, the term in x^2 corresponds to a linear
error in the spacing. In like manner, the term in y^2 corresponds to a
general _curvature_ of the lines (fig. 10), and does not influence the
definition at the (primary) focus, although it may introduce
astigmatism.[8] If we suppose that everything is symmetrical on the
two sides of the primary plane y = 0, the coefficients B, [beta],
[delta] vanish. In spite of any inequality between [rho] and [rho]',
the definition will be good to this order of approximation, provided
[alpha] and [gamma] vanish. The former measures the _thickness_ of the
primary focal line, and the latter measures its _curvature_. The error
of ruling giving rise to [alpha] is one in which the intervals
increase or decrease in _both_ directions from the centre outwards
(fig. 11), and it may often be compensated by a slight rotation in
azimuth of the object-glass of the observing telescope. The term in
[gamma] corresponds to a _variation_ of curvature in crossing the
grating (fig. 12).
When the plane zx is not a plane of symmetry, we have to consider the
terms in xy, x^2y, and y^3. The first of these corresponds to a
deviation from parallelism, causing the interval to alter gradually as
we pass _along_ the lines (fig. 13). The error thus arising may be
compensated by a rotation of the object-glass about one of the
diameters y = [+-] x. The term in x^2y corresponds to a deviation from
parallelism in the same direction on both sides of the central line
(fig. 14); and that in y^3 would be caused by a curvature such that
there is a point of inflection at the middle of each line (fig. 15).
All the errors, except that depending on [alpha], and especially those
depending on [gamma] and [delta], can be diminished, without loss of
resolving power, by contracting the _vertical_ aperture. A linear
error in the spacing, and a general curvature of the lines, are
eliminated in the ordinary use of a grating.
The explanation of the difference of focus upon the two sides as due
to unequal spacing was verified by Cornu upon gratings purposely
constructed with an increasing interval. He has also shown how to rule
a plane surface with lines so disposed that the grating shall of
itself give well-focused spectra.
[Illustration: FIG. 16.]
A similar idea appears to have guided H. A. Rowland to his brilliant
invention of concave gratings, by which spectra can be photographed
without any further optical appliance. In these instruments the lines
are ruled upon a spherical surface of speculum metal, and mark the
intersections of the surface by a system of parallel and equidistant
planes, of which the middle member passes through the centre of the
sphere. If we consider for the present only the primary plane of
symmetry, the figure is reduced to two dimensions. Let AP (fig. 16)
represent the surface of the grating, O being the centre of the
circle. Then, if Q be any radiant point and Q' its image (primary
focus) in the spherical mirror AP, we have
1 1 2cos[phi]
-- + - = ---------,
v1 u a
where v1 = AQ', u = AQ, a = OA, [phi] = angle of incidence QAO, equal
to the angle of reflection Q'AO. If Q be on the circle described upon
OA as diameter, so that u = a cos [phi], then Q' lies also upon the
same circle; and in this case it follows from the symmetry that the
unsymmetrical aberration (depending upon a) vanishes.
This disposition is adopted in Rowland's instrument; only, in addition
to the central image formed at the angle [phi]' = [phi], there are a
series of spectra with various values of [phi]', but all disposed upon
the same circle. Rowland's investigation is contained in the paper
already referred to; but the following account of the theory is in the
form adopted by R. T. Glazebrook (_Phil. Mag._, 1883).
In order to find the difference of optical distances between the
courses QAQ', QPQ', we have to express QP - QA, PQ' - AQ'. To find the
former, we have, if OAQ = [phi], AOP = [omega],
QP^2 = u^2 + 4a^2sin^21/2[omega] - 4au sin 1/2[omega] sin (1/2[omega] - [phi])
= (u + a sin[phi] sin[omega])^2 - a^2 sin^2[phi] sin^2[omega] + 4a sin^2 1/2[omega](a - u cos[phi]).