Chapter XVII: Part 17

The theory was further developed by A. J. Fresnel (1815), who gave a formula equivalent to (5) below. But it is to J. von Fraunhofer that we owe most of our knowledge upon this subject. His recent discovery of the "fixed lines" allowed a precision of observation previously impossible. He constructed gratings up to 340 periods to the inch by straining fine wire over screws. Subsequently he ruled gratings on a layer of gold-leaf attached to glass, or on a layer of grease similarly supported, and again by attacking the glass itself with a diamond point. The best gratings were obtained by the last method, but a suitable diamond point was hard to find, and to preserve. Observing through a telescope with light perpendicularly incident, he showed that the position of any ray was dependent only upon the grating interval, viz. the distance from the centre of one wire or line to the centre of the next, and not otherwise upon the thickness of the wire and the magnitude of the interspace. In different gratings the lengths of the spectra and their distances from the axis were inversely proportional to the grating interval, while with a given grating the distances of the various spectra from the axis were as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of wave-lengths, and the method of separating the overlapping spectra by a prism dispersing in the perpendicular direction. He described also the complicated patterns seen when a point of light is viewed through two superposed gratings, whose lines cross one another perpendicularly or obliquely. The above observations relate to transmitted light, but Fraunhofer extended his inquiry to the light _reflected_. To eliminate the light returned from the hinder surface of an engraved grating, he covered it with a black varnish. It then appeared that under certain angles of incidence parts of the resulting spectra were _completely polarized_. These remarkable researches of Fraunhofer, carried out in the years 1817-1823, are republished in his _Collected Writings_ (Munich, 1888).

The principle underlying the action of gratings is identical with that
discussed in S 2, and exemplified in J. L. Soret's "zone plates." The
alternate Fresnel's zones are blocked out or otherwise modified; in
this way the original compensation is upset and a revival of light
occurs in unusual directions. If the source be a point or a line, and
a collimating lens be used, the incident waves may be regarded as
plane. If, further, on leaving the grating the light be received by a
focusing lens, e.g. the object-glass of a telescope, the Fresnel's
zones are reduced to parallel and equidistant straight strips, which
at certain angles coincide with the ruling. The directions of the
lateral spectra are such that the passage from one element of the
grating to the corresponding point of the next implies a retardation
of an integral number of wave-lengths. If the grating be composed of
alternate transparent and opaque parts, the question may be treated by
means of the general integrals (S 3) by merely limiting the
integration to the transparent parts of the aperture. For an
investigation upon these lines the reader is referred to Airy's
_Tracts_, to Verdet's _Lecons_, or to R. W. Wood's _Physical Optics_.
If, however, we assume the theory of a simple rectangular aperture (S
3); the results of the ruling can be inferred by elementary methods,
which are perhaps more instructive.

Apart from the ruling, we know that the image of a mathematical line
will be a series of narrow bands, of which the central one is by far
the brightest. At the middle of this band there is complete agreement
of phase among the secondary waves. The dark lines which separate the
bands are the places at which the phases of the secondary wave range
over an integral number of periods. If now we suppose the aperture AB
to be covered by a great number of opaque strips or bars of width d,
separated by transparent intervals of width a, the condition of things
in the directions just spoken of is not materially changed. At the
central point there is still complete agreement of phase; but the
amplitude is diminished in the ratio of a : a + d. In another
direction, making a small angle with the last, such that the
projection of AB upon it amounts to a few wave-lengths, it is easy to
see that the mode of interference is the same as if there were no
ruling. For example, when the direction is such that the projection of
AB upon it amounts to one wave-length, the elementary components
neutralize one another, because their phases are distributed
symmetrically, though discontinuously, round the entire period. The
only effect of the ruling is to diminish the amplitude in the ratio a
: a + d; and, except for the difference in illumination, the
appearance of a line of light is the same as if the aperture were
perfectly free.

The lateral (spectral) images occur in such directions that the
projection of the element (a + d) of the grating upon them is an exact
multiple of [lambda]. The effect of each of the n elements of the
grating is then the same; and, unless this vanishes on account of a
particular adjustment of the ratio a : d, the resultant amplitude
becomes comparatively very great. These directions, in which the
retardation between A and B is exactly mn[lambda], may be called the
principal directions. On either side of any one of them the
illumination is distributed according to the same law as for the
central image (m = 0), vanishing, for example, when the retardation
amounts to (mn [+-] 1)[lambda]. In considering the relative brightnesses
of the different spectra, it is therefore sufficient to attend merely
to the principal directions, provided that the whole deviation be not
so great that its cosine differs considerably from unity.

We have now to consider the amplitude due to a single element, which
we may conveniently regard as composed of a transparent part a bounded
by two opaque parts of width 1/2d. The phase of the resultant effect is
by symmetry that of the component which comes from the middle of a.
The fact that the other components have phases differing from this by
amounts ranging between [+-] am[pi]/(a + d) causes the resultant
amplitude to be less than for the central image (where there is
complete phase agreement). If Bm denote the brightness of the m^th
lateral image, and B0 that of the central image, we have

_ _+ am[pi]/(a + d) _
| / 2am[pi] |^2 /a + d \^2 am[pi]
B_m : B0 = | | cosx dx :- ------- | = ( ------ ) sin^2 ------ (1).
|_ _/ a + d _| \am[pi]/ a + d
-am[pi]/(a + d)

If B denotes the brightness of the central image when the whole of the
space occupied by the grating is transparent, we have

B0 : B = a^2 : (a + d)^2,

and thus

1 am[pi]
Bm : B = --------- sin^2 ------ (2).
m^2[pi]^2 a + d

The sine of an angle can never be greater than unity; and consequently
under the most favourable circumstances only 1/m^2[pi]^2 of the original
light can be obtained in the m^th spectrum. We conclude that, with a
grating composed of transparent and opaque parts, the utmost light
obtainable in any one spectrum is in the first, and there amounts to
1/[pi]^2, or about 1/10, and that for this purpose a and d must be
equal. When d = a the general formula becomes

sin^2 1/2m[pi]
Bm : B = ------------- (3),
m^2[pi]^2

showing that, when m is even, Bm vanishes, and that, when m is odd,

Bm : B = 1/m^2[pi]^2.

The third spectrum has thus only 1/9 of the brilliancy of the first.

Another particular case of interest is obtained by supposing a small
relatively to (a + d). Unless the spectrum be of very high order, we
have simply

Bm : B = a/(a + d)^2 (4);

so that the brightnesses of all the spectra are the same.

The light stopped by the opaque parts of the grating, together with
that distributed in the central image and lateral spectra, ought to
make up the brightness that would be found in the central image, were
all the apertures transparent. Thus, if a = d, we should have

1 1 2 / 1 1 \
1 = - + - + ------ ( 1 + - + -- + ... ),
2 4 [pi]^2 \ 9 25 /

which is true by a known theorem. In the general case

___m=[oo]
a / a \^2 2 \ 1 /m[pi]a\
----- = ( ----- ) + ------ > -- sin^2( ------ ),
a + d \a + d/ [pi]^2 /__ m^2 \ a + d/
m=1

a formula which may be verified by Fourier's theorem.

According to a general principle formulated by J. Babinet, the
brightness of a lateral spectrum is not affected by an interchange of
the transparent and opaque parts of the grating. The vibrations
corresponding to the two parts are precisely antagonistic, since if
both were operative the resultant would be zero. So far as the
application to gratings is concerned, the same conclusion may be
derived from (2).

From the value of Bm : B0 we see that no lateral spectrum can surpass
the central image in brightness; but this result depends upon the
hypothesis that the ruling acts by opacity, which is generally very
far from being the case in practice. In an engraved glass grating
there is no opaque material present by which light could be absorbed,
and the effect depends upon a difference of retardation in passing the
alternate parts. It is possible to prepare gratings which give a
lateral spectrum brighter than the central image, and the explanation
is easy. For if the alternate parts were equal and alike transparent,
but so constituted as to give a relative retardation of 1/2[lambda], it
is evident that the central image would be entirely extinguished,
while the first spectrum would be four times as bright as if the
alternate parts were opaque. If it were possible to introduce at every
part of the aperture of the grating an arbitrary retardation, all the
light might be concentrated in any desired spectrum. By supposing the
retardation to vary uniformly and continuously we fall upon the case
of an ordinary prism: but there is then no diffraction spectrum in the
usual sense. To obtain such it would be necessary that the retardation
should gradually alter by a wave-length in passing over any element of
the grating, and then fall back to its previous value, thus springing
suddenly over a wave-length (_Phil. Mag._, 1874, 47, p. 193). It is
not likely that such a result will ever be fully attained in practice;
but the case is worth stating, in order to show that there is no
theoretical limit to the concentration of light of assigned
wave-length in one spectrum, and as illustrating the frequently
observed unsymmetrical character of the spectra on the two sides of
the central image.[4]

We have hitherto supposed that the light is incident perpendicularly
upon the grating; but the theory is easily extended. If the incident
rays make an angle [theta] with the normal (fig. 6), and the
diffracted rays make an angle [phi] (upon the same side), the relative
retardation from each element of width (a + d) to the next is (a + d)
(sin[theta] + sin[phi]); and this is the quantity which is to be
equated to m[lambda]. Thus

sin[theta] + sin[phi] = 2 sin 1/2([theta] + [phi]) cos 1/2([theta] - [phi]) = m[lambda]/(a + d) (5).

The "deviation" is ([theta] + [phi]), and is therefore a minimum when
[theta] = [phi], i.e. when the grating is so situated that the angles
of incidence and diffraction are equal.

In the case of a reflection grating the same method applies. If
[theta] and [phi] denote the angles with the normal made by the
incident and diffracted rays, the formula (5) still holds, and, if the
deviation be reckoned from the direction of the regularly reflected
rays, it is expressed as before by ([theta] + [phi]), and is a minimum
when [theta] = [phi], that is, when the diffracted rays return upon
the course of the incident rays.

In either case (as also with a prism) the position of minimum
deviation leaves the width of the beam unaltered, i.e. neither
magnifies nor diminishes the angular width of the object under view.

From (5) we see that, when the light falls perpendicularly upon a
grating ([theta] = 0), there is no spectrum formed (the image
corresponding to m = 0 not being counted as a spectrum), if the
grating interval [sigma] or (a + d) is less than [lambda]. Under these
circumstances, if the material of the grating be completely
transparent, the whole of the light must appear in the direct image,
and the ruling is not perceptible. From the absence of spectra
Fraunhofer argued that there must be a microscopic limit represented
by [lambda]; and the inference is plausible, to say the least (_Phil.
Mag._, 1886). Fraunhofer should, however, have fixed the microscopic
limit at 1/2[lambda], as appears from (5), when we suppose [theta] =
1/2[pi], [phi] = 1/2[pi].

We will now consider the important subject of the resolving power of
gratings, as dependent upon the number of lines (n) and the order of
the spectrum observed (m). Let BP (fig. 8) be the direction of the
principal maximum (middle of central band) for the wave-length
[lambda] in the m^th spectrum. Then the relative retardation of the
extreme rays (corresponding to the edges A, B of the grating) is
mn[lambda]. If BQ be the direction for the first minimum (the darkness
between the central and first lateral band), the relative retardation
of the extreme rays is (mn + 1)[lambda]. Suppose now that [lambda] +
[delta][lambda] is the wave-length for which BQ gives the principal
maximum, then

(mn + 1)[lambda] = mn([lambda] + [delta][lambda]);

whence

[delta][lambda]/[lambda] = 1/mn (6).

According to our former standard, this gives the smallest difference
of wave-lengths in a double line which can be just resolved; and we
conclude that the resolving power of a grating depends only upon the
total number of lines, and upon the order of the spectrum, without
regard to any other considerations. It is here of course assumed that
the n lines are really utilized.

In the case of the D lines the value of [delta][lambda]/[lambda] is
about 1/1000; so that to resolve this double line in the first
spectrum requires 1000 lines, in the second spectrum 500, and so on.

It is especially to be noticed that the resolving power does not
depend directly upon the closeness of the ruling. Let us take the case
of a grating 1 in. broad, and containing 1000 lines, and consider the
effect of interpolating an additional 1000 lines, so as to bisect the
former intervals. There will be destruction by interference of the
first, third and odd spectra generally; while the advantage gained in
the spectra of even order is not in dispersion, nor in resolving
power, but simply in brilliancy, which is increased four times. If we
now suppose half the grating cut away, so as to leave 1000 lines in
half an inch, the dispersion will not be altered, while the brightness
and resolving power are halved.

There is clearly no theoretical limit to the resolving power of
gratings, even in spectra of given order. But it is possible that, as
suggested by Rowland,[5] the structure of natural spectra may be too
coarse to give opportunity for resolving powers much higher than those
now in use. However this may be, it would always be possible, with the
aid of a grating of given resolving power, to construct artificially
from white light mixtures of slightly different wave-length whose
resolution or otherwise would discriminate between powers inferior and
superior to the given one.[6]

If we define as the "dispersion" in a particular part of the spectrum
the ratio of the angular interval d[theta] to the corresponding
increment of wave-length d[lambda], we may express it by a very simple
formula. For the alteration of wave-length entails, at the two limits
of a diffracted wave-front, a relative retardation equal to
mnd[lambda]. Hence, if a be the width of the diffracted beam, and
d[theta] the angle through which the wave-front is turned,

ad[theta] = mn d[lambda],

or dispersion = mn/a (7).

The resolving power and the width of the emergent beam fix the optical
character of the instrument. The latter element must eventually be
decreased until less than the diameter of the pupil of the eye. Hence
a wide beam demands treatment with further apparatus (usually a
telescope) of high magnifying power.

In the above discussion it has been supposed that the ruling is
accurate, and we have seen that by increase of m a high resolving
power is attainable with a moderate number of lines. But this
procedure (apart from the question of illumination) is open to the
objection that it makes excessive demands upon accuracy. According to
the principle already laid down it can make but little difference in
the principal direction corresponding to the first spectrum, provided
each line lie within a quarter of an interval (a + d) from its
theoretical position. But, to obtain an equally good result in the
m^th spectrum, the error must be less than 1/m of the above amount.[7]

There are certain errors of a systematic character which demand
special consideration. The spacing is usually effected by means of a
screw, to each revolution of which corresponds a large number (e.g.
one hundred) of lines. In this way it may happen that although there
is almost perfect periodicity with each revolution of the screw after
(say) 100 lines, yet the 100 lines themselves are not equally spaced.
The "ghosts" thus arising were first described by G. H. Quincke
(_Pogg. Ann._, 1872, 146, p. 1), and have been elaborately
investigated by C. S. Peirce (_Ann. Journ. Math._, 1879, 2, p. 330),
both theoretically and experimentally. The general nature of the
effects to be expected in such a case may be made clear by means of an
illustration already employed for another purpose. Suppose two similar
and accurately ruled transparent gratings to be superposed in such a
manner that the lines are parallel. If the one set of lines exactly
bisect the intervals between the others, the grating interval is
practically halved, and the previously existing spectra of odd order
vanish. But a very slight relative displacement will cause the
apparition of the odd spectra. In this case there is approximate
periodicity in the half interval, but complete periodicity only after
the whole interval. The advantage of approximate bisection lies in the
superior brilliancy of the surviving spectra; but in any case the
compound grating may be considered to be perfect in the longer
interval, and the definition is as good as if the bisection were
accurate.

| | | | | ( ( ( | | | | ) | (

FIG. 9.--x^2. FIG. 10.--y^2. FIG. 11.--x^3. FIG. 12.--xy^2.

/ / /
\ | | / | \ | | | |
/ / /

FIG. 13.--xy. FIG. 14.--x^2y. FIG. 15.--y^3.]

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.

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]).

Now as far as [omega]^4

4 sin^2 1/2[omega] = sin^2[omega] + 1/4sin^4[omega],

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

I = u^2 + v^2 + 2uv cos[alpha],

which becomes on putting for u, v, and [alpha] their values, and
putting

/ f k[eta]l \^2
( ------- sin ------- ) = Q (3),
\k[eta]l f /
_ _
4l^2 [pi][xi]h | / 2[pi]R 2[pi][xi]h\ |
I = Q . ------------ sin^2 --------- |2 + 2 cos ( -------- - ---------- ) | (4).
[pi]^2[xi]^2 [lambda]f |_ \[lambda] [lambda]f / _|

If the subject of examination be a luminous line parallel to [eta], we
shall obtain what we require by integrating (4) with respect to [eta]
from -[oo] to +[oo]. The constant multiplier is of no especial
interest so that we may take as applicable to the image of a line
_ _
2 [pi][xi]h | / 2[pi]R 2[pi][xi]h \ |
I = ------ sin^2 --------- |1 + cos ( -------- - ---------- ) | (5).
[xi]^2 [lambda]f |_ \[lambda] [lambda]f / _|

If R = 1/2[lambda], I vanishes at [xi]= 0; but the whole illumination,
represented by
_
/ +[oo]
| I d[xi], is independent of the value of R. If R = 0,
_/-[oo]

1 2[pi][xi]h
I = ------ sin^2 ----------,
[xi]^2 [lambda]f

in agreement with S 3, where a has the meaning here attached to 2h.

The expression (5) gives the illumination at [xi] due to that part of
the complete image whose geometrical focus is at [xi] = 0, the
retardation for this component being R. Since we have now to integrate
for the whole illumination at a particular point O due to all the
components which have their foci in its neighbourhood, we may
conveniently regard O as origin. [xi] is then the co-ordinate
relatively to O of any focal point O' for which the retardation is R;
and the required result is obtained by simply integrating (5) with
respect to [xi] from -[oo] to +[oo]. To each value of [xi] corresponds
a different value of [lambda], and (in consequence of the dispersing
power of the plate) of R. The variation of [lambda] may, however, be
neglected in the integration, except in 2[pi]R/[lambda], where a small
variation of [lambda] entails a comparatively large alteration of
phase. If we write

[rho] = 2[pi]R/[lambda] (6),

we must regard [rho] as a function of [xi], and we may take with
sufficient approximation under any ordinary circumstances

[rho] = [rho]' + [=omega][xi] (7),

where [rho]' denotes the value of [rho] at O, and [=omega] is a
constant, which is positive when the retarding plate is held at the
side on which the lue of the spectrum _is seen_. The possibility of
dark bands depends upon [=omega] being positive. Only in this case can

cos {[rho]' + ([=omega] - 2[pi]h/[lambda]f)[xi]}

retain the constant value -1 throughout the integration, and then only
when

[=omega] = 2[pi]h / [lambda]f (8)

and

cos [rho]' = -1 (9).

The first of these equations is the condition for the formation of
dark bands, and the second marks their situation, which is the same
as that determined by the imperfect theory.

The integration can be effected without much difficulty. For the first
term in (5) the evaluation is effected at once by a known formula. In
the second term if we observe that

cos {[rho]' +([=omega] - 2[pi]h/[lambda]f)[xi]} = cos {[rho]'- g1[xi]}
= cos [rho]' cos g1[xi] + sin [rho]' sin g1[xi],

we see that the second part vanishes when integrated, and that the
remaining integral is of the form

_+[oo]
/ d[xi]
w = | sin^2 h1[xi] cos g1[xi] ------,
_/-[oo] [xi]^2

where

h1 = [pi]h/[lambda]f, g1 = [omega] - 2[pi]h/[lambda]f (10).

By differentiation with respect to g1 it may be proved that

w = 0 from g1 = -[oo] to g1 = -2h1,
w = 1/2[pi](2h1 + g1) from g1 = -2h1 to g1 = 0,
w = 1/2[pi](2h1 - g1) from g1 = 0 to g1 = 2h1,
w = 0 from g1 = 2h1 to g1 = [oo].

The integrated intensity, I', or

2[pi]h1 + 2 cos[rho]w,

is thus

I' = 2[pi]h1 (11),

when g1 numerically exceeds 2h1; and, when g1 lies between [+-]2h1,

I = [pi]2h1 + (2h1 - [sqrt] g1^2) cos[rho]' (12).

It appears therefore that there are no bands at all unless [omega]
lies between 0 and +4h1, and that within these limits the best bands
are formed at the middle of the range when [omega] = 2h1. The
formation of bands thus requires that the retarding plate be held upon
the side already specified, so that [omega] be positive; and that the
thickness of the plate (to which [omega] is proportional) do not
exceed a certain limit, which we may call 2T0. At the best thickness
T0 the bands are black, and not otherwise.

The linear width of the band (e) is the increment of [xi] which alters
[rho] by 2[pi], so that

e = 2[pi]/[=omega] (13).

With the best thickness

[=omega] = 2[pi]h/[lambda]f (14),

so that in this case

e = [lambda]f/h (15).

The bands are thus of the same width as those due to two infinitely
narrow apertures coincident with the central lines of the retarded and
unretarded streams, the subject of examination being itself a fine
luminous line.

If it be desired to see a given number of bands in the whole or in any
part of the spectrum, the thickness of the retarding plate is thereby
determined, independently of all other considerations. But in order
that the bands may be really visible, and still more in order that
they may be black, another condition must be satisfied. It is
necessary that the aperture of the pupil be accommodated to the
angular extent of the spectrum, or reciprocally. Black bands will be
too fine to be well seen unless the aperture (2h) of the pupil be
somewhat contracted. One-twentieth to one-fiftieth of an inch is
suitable. The aperture and the number of bands being both fixed, the
condition of blackness determines the angular magnitude of a band and
of the spectrum. The use of a grating is very convenient, for not only
are there several spectra in view at the same time, but the dispersion
can be varied continuously by sloping the grating. The slits may be
cut out of tin-plate, and half covered by mica or "microscopic glass,"
held in position by a little cement.

If a telescope be employed there is a distinction to be observed,
according as the half-covered aperture is between the eye and the
ocular, or in front of the object-glass. In the former case the
function of the telescope is simply to increase the dispersion, and
the formation of the bands is of course independent of the particular
manner in which the dispersion arises. If, however, the half-covered
aperture be in front of the object-glass, the phenomenon is magnified
as a whole, and the desirable relation between the (unmagnified)
dispersion and the aperture is the same as without the telescope.
There appears to be no further advantage in the use of a telescope
than the increased facility of accommodation, and for this of course a
very low power suffices.

The original investigation of Stokes, here briefly sketched, extends
also to the case where the streams are of unequal width h, k, and are
separated by an interval 2g. In the case of unequal width the bands
cannot be black; but if h = k, the finiteness of 2g does not preclude
the formation of black bands.

The theory of Talbot's bands with a half-covered _circular_ aperture
has been considered by H. Struve (_St Peters. Trans._, 1883, 31, No.
1).

The subject of "Talbot's bands" has been treated in a very instructive
manner by A. Schuster (_Phil. Mag._, 1904), whose point of view offers
the great advantage of affording an instantaneous explanation of the
peculiarity noticed by Brewster. A plane _pulse_, i.e. a disturbance
limited to an infinitely thin slice of the medium, is supposed to fall
upon a parallel grating, which again may be regarded as formed of
infinitely thin wires, or infinitely narrow lines traced upon glass.
The secondary pulses diverted by the ruling fall upon an object-glass
as usual, and on arrival at the focus constitute a procession equally
spaced in time, the interval between consecutive members depending
upon the obliquity. If a retarding plate be now inserted so as to
operate upon the pulses which come from one side of the grating, while
leaving the remainder unaffected, we have to consider what happens at
the focal point chosen. A full discussion would call for the formal
application of Fourier's theorem, but some conclusions of importance
are almost obvious.

Previously to the introduction of the plate we have an effect
corresponding to wave-lengths closely grouped around the principal
wave-length, viz. [sigma] sin [phi], where [sigma] is the
grating-interval and [phi] the obliquity, the closeness of the
grouping increasing with the number of intervals. In addition to these
wave-lengths there are other groups centred round the wave-lengths
which are submultiples of the principal one--the overlapping spectra
of the second and higher orders. Suppose now that the plate is
introduced so as to cover naif the aperture and that it retards those
pulses which would otherwise arrive first. The consequences must
depend upon the amount of the retardation. As this increases from
zero, the two processions which correspond to the two halves of the
aperture begin to overlap, and the overlapping gradually increases
until there is almost complete superposition. The stage upon which we
will fix our attention is that where the one procession bisects the
intervals between the other, so that a new simple procession is
constituted, containing the same number of members as before the
insertion of the plate, but now spaced at intervals only half as
great. It is evident that the effect at the focal point is the
obliteration of the first and other spectra of odd order, so that as
regards the spectrum of the first order we may consider that the two
beams _interfere_. The formation of black bands is thus explained, and
it requires that the plate be introduced upon one particular side, and
that the amount of the retardation be adjusted to a particular value.
If the retardation be too little, the overlapping of the processions
is incomplete, so that besides the procession of half period there are
residues of the original processions of full period. The same thing
occurs if the retardation be too great. If it exceed the double of the
value necessary for black bands, there is again no overlapping and
consequently no interference. If the plate be introduced upon the
other side, so as to retard the procession originally in arrear, there
is no overlapping, whatever may be the amount of retardation. In this
way the principal features of the phenomenon are accounted for, and
Schuster has shown further how to extend the results to spectra having
their origin in prisms instead of gratings.

10. _Diffraction when the Source of Light is not seen in Focus._--The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail; but in view of their historical interest and of the ease with which many of the experiments may be tried, some account of their theory cannot be omitted. One or two examples have already attracted our attention when considering Fresnel's zones, viz. the shadow of a circular disk and of a screen circularly perforated.

Fresnel commenced his researches with an examination of the fringes, external and internal, which accompany the shadow of a narrow opaque strip, such as a wire. As a source of light he used sunshine passing through a very small hole perforated in a metal plate, or condensed by a lens of short focus. In the absence of a heliostat the latter was the more convenient. Following, unknown to himself, in the footsteps of Young, he deduced the principle of interference from the circumstance that the darkness of the interior bands requires the co-operation of light from both sides of the obstacle. At first, too, he followed Young in the view that the exterior bands are the result of interference between the direct light and that reflected from the edge of the obstacle, but he soon discovered that the character of the edge--e.g. whether it was the cutting edge or the back of a razor--made no material difference, and was thus led to the conclusion that the explanation of these phenomena requires nothing more than the application of Huygens's principle to the unobstructed parts of the wave. In observing the bands he received them at first upon a screen of finely ground glass, upon which a magnifying lens was focused; but it soon appeared that the ground glass could be dispensed with, the diffraction pattern being viewed in the same way as the image formed by the object-glass of a telescope is viewed through the eye-piece. This simplification was attended by a great saving of light, allowing measures to be taken such as would otherwise have presented great difficulties.

In theoretical investigations these problems are usually treated as of
two dimensions only, everything being referred to the plane passing
through the luminous point and perpendicular to the diffracting edges,
supposed to be straight and parallel. In strictness this idea is
appropriate only when the source is a luminous line, emitting
cylindrical waves, such as might be obtained from a luminous point
with the aid of a cylindrical lens. When, in order to apply Huygens's
principle, the wave is supposed to be broken up, the phase is the same
at every element of the surface of resolution which lies upon a line
perpendicular to the plane of reference, and thus the effect of the
whole line, or rather infinitesimal strip, is related in a constant
manner to that of the element which lies in the plane of reference,
and may be considered to be represented thereby. The same method of
representation is applicable to spherical waves, issuing from a
_point_, if the radius of curvature be large; for, although there is
variation of phase along the length of the infinitesimal strip, the
whole effect depends practically upon that of the central parts where
the phase is sensibly constant.[10]

In fig. 17 APQ is the arc of the circle representative of the
wave-front of resolution, the centre being at O, and the radius QA
being equal to a. B is the point at which the effect is required,
distant a + b from O, so that AB = b, AP = s, PQ = ds.

Taking as the standard phase that of the secondary wave from A, we may
represent the effect of PQ by

/t [delta] \
cos 2[pi] ( - - -------- ).ds,
\r [lambda]/

where [delta] = BP - AP is the retardation at B of the wave from P
relatively to that from A.

Now

[delta] = (a + b) s^2/2ab (1),

so that, if we write

2[pi][delta] = [pi](a + b)s^2 [pi]v^2
------------ --------------- = ------ (2),
[lambda] ab[lambda] 2

the effect at B is
_ _
/ab[lambda]\1/2 / 2[pi]t / 2[pi]t / \
( ---------- ) ( cos ------ | cos 1/2[pi]v^2.dv + sin ------ | sin 1/2[pi]v^2.dv ) (3),
\2(a + b) / \ [tau] _/ [tau] _/ /

the limits of integration depending upon the disposition of the
diffracting edges. When a, b, [lambda] are regarded as constant, the
first factor may be omitted,--as indeed should be done for
consistency's sake, inasmuch as other factors of the same nature have
been omitted already.

The intensity I^2, the quantity with which we are principally
concerned, may thus be expressed

_ _
/ / \^2 / / \^2
I^2= ( | cos 1/2[pi]v^2.dv ) + ( | sin 1/2[pi]v^2.dv ) (4).
\ _/ / \ _/ /

These integrals, taken from v = 0, are known as Fresnel's integrals;
we will denote them by C and S, so that
_ _
/ v / v
C = | cos 1/2[pi]v^2.dv, S = | cos 1/2[pi]v^2.dv (5).
_/0 _/0

When the upper limit is infinity, so that the limits correspond to the
inclusion of half the primary wave, C and S are both equal to 1/2, by a
known formula; and on account of the rapid fluctuation of sign the
parts of the range beyond very moderate values of v contribute but
little to the result.

Ascending series for C and S were given by K. W. Knockenhauer, and are
readily investigated. Integrating by parts, we find

_v _v
/ i.1/2[pi]v^2 i.1/2[pi]v^2 1 / i.1/2[pi]v^2
C + iS = | e dv = e . v - - i[pi] | e dv^3;
_/0 3 _/0

and, by continuing this process,