f1 = 800 inches.
The image of the sun thrown upon a screen at a distance exceeding 66
ft., through a hole 1/5 in. in diameter, is therefore at least as well
defined as that seen direct.
As the minimum focal length increases with the square of the aperture,
a quite impracticable distance would be required to rival the
resolving power of a modern telescope. Even for an aperture of 4 in.,
f1 would have to be 5 miles.
A similar argument may be applied to find at what point an achromatic
lens becomes sensibly superior to a single one. The question is
whether, when the adjustment of focus is correct for the central rays
of the spectrum, the error of phase for the most extreme rays (which
it is necessary to consider) amounts to a quarter of a wave-length. If
not, the substitution of an achromatic lens will be of no advantage.
Calculation shows that, if the aperture be 1/5 in., an achromatic lens
has no sensible advantage if the focal length be greater than about 11
in. If we suppose the focal length to be 66 ft., a single lens is
practically perfect up to an aperture of 1.7 in.
Another obvious inference from the necessary imperfection of optical
images is the uselessness of attempting anything like an absolute
destruction of spherical aberration. An admissible error of phase of
1/4[lambda] will correspond to an error of 1/8[lambda] in a reflecting
and 1/2[lambda] in a (glass) refracting surface, the incidence in both
cases being perpendicular. If we inquire what is the greatest
admissible longitudinal aberration ([delta]f) in an object-glass
according to the above rule, we find
[delta]f = [lambda][alpha]^-2 (2),
[alpha] being the angular semi-aperture.
In the case of a single lens of glass with the most favourable
curvatures, [delta]f is about equal to [alpha]^2f, so that [alpha]^4
must not exceed [lambda]/f. For a lens of 3 ft. focus this condition
is satisfied if the aperture does not exceed 2 in.
When parallel rays fall directly upon a spherical mirror the
longitudinal aberration is only about one-eighth as great as for the
most favourably shaped single lens of equal focal length and aperture.
Hence a spherical mirror of 3 ft. focus might have an aperture of 21/2
in., and the image would not suffer materially from aberration.
On the same principle we may estimate the least visible displacement
of the eye-piece of a telescope focused upon a distant object, a
question of interest in connexion with range-finders. It appears
(_Phil. Mag._, 1885, 20, p. 354) that a displacement [delta]f from the
true focus will not sensibly impair definition, provided
[delta]f < f^2[lambda]/R^2 (3),
2R being the diameter of aperture. The linear accuracy required is
thus a function of the _ratio_ of aperture to focal length. The
formula agrees well with experiment.
The principle gives an instantaneous solution of the question of the
ultimate optical efficiency in the method of "mirror-reading," as
commonly practised in various physical observations. A rotation by
which one edge of the mirror advances 1/4[lambda] (while the other edge
retreats to a like amount) introduces a phase-discrepancy of a whole
period where before the rotation there was complete agreement. A
rotation of this amount should therefore be easily visible, but the
limits of resolving power are being approached; and the conclusion is
independent of the focal length of the mirror, and of the employment
of a telescope, provided of course that the reflected image is seen in
focus, and that the full width of the mirror is utilized.
A comparison with the method of a material pointer, attached to the
parts whose rotation is under observation, and viewed through a
microscope, is of interest. The limiting efficiency of the microscope
is attained when the angular aperture amounts to 180 deg.; and it is
evident that a lateral displacement of the point under observation
through 1/2[lambda] entails (at the old image) a phase-discrepancy of a
whole period, one extreme ray being accelerated and the other retarded
by half that amount. We may infer that the limits of efficiency in the
two methods are the same when the length of the pointer is equal to
the width of the mirror.
[Illustraton: FIG. 5.]
We have seen that in perpendicular reflection a surface error not
exceeding 1/8[lambda] may be admissible. In the case of oblique
reflection at an angle [phi], the error of retardation due to an
elevation BD (fig. 5) is
QQ' - QS = BD sec [phi](1 - cos SQQ') = BD sec [phi] (1 + cos 2[phi]) = 2BD cos [phi];
from which it follows that an error of given magnitude in the figure
of a surface is less important in oblique than in perpendicular
reflection. It must, however, be borne in mind that errors can
sometimes be compensated by altering adjustments. If a surface
intended to be flat is affected with a slight general curvature, a
remedy may be found in an alteration of focus, and the remedy is the
less complete as the reflection is more oblique.
The formula expressing the optical power of prismatic spectroscopes
may readily be investigated upon the principles of the wave theory.
Let A0B0 be a plane wave-surface of the light before it falls upon the
prisms, AB the corresponding wave-surface for a particular part of the
spectrum after the light has passed the prisms, or after it has passed
the eye-piece of the observing telescope. The path of a ray from the
wave-surface A0B0 to A or B is determined by the condition that the
optical distance, [int] [mu]ds, is a minimum; and, as AB is by
supposition a wave-surface, this optical distance is the same for both
points. Thus
_ _
/ /
| [mu]ds (for A) = | [mu]ds (for B) (4).
_/ _/
We have now to consider the behaviour of light belonging to a
neighbouring part of the spectrum. The path of a ray from the
wave-surface A0B0 to the point A is changed; but in virtue of the
minimum property the change may be neglected in calculating the
optical distance, as it influences the result by quantities of the
second order only in the changes of refrangibility. Accordingly, the
optical distance from A0B0 to A is represented by [int]([mu] +
[delta][mu])ds, the integration being along the original path A0 ...
A; and similarly the optical distance between A0B0 and B is
represented by [int] ([mu] + [delta][mu])ds, the integration being
along B0 ... B. In virtue of (4) the difference of the optical
distances to A and B is
_ _
/ /
| [delta][mu]ds (along B0 ... B) - | [delta][mu]ds (along A0 ... A) (5).
_/ _/
The new wave-surface is formed in such a position that the optical
distance is constant; and therefore the _dispersion_, or the angle
through which the wave-surface is turned by the change of
refrangibility, is found simply by dividing (5) by the distance AB.
If, as in common flint-glass spectroscopes, there is only one
dispersing substance, [int] [delta][mu] ds = [delta][mu].s, where s is
simply the thickness traversed by the ray. If t2 and t1 be the
thicknesses traversed by the extreme rays, and a denote the width of
the emergent beam, the dispersion [theta] is given by
[theta] = [delta][mu](t2 - t1)/a,
or, if t1 be negligible,
[theta] = [delta][mu]t/a (6).
The condition of resolution of a double line whose components subtend
an angle [theta] is that [theta] must exceed [lambda]/a. Hence, in
order that a double line may be resolved whose components have indices
[mu] and [mu] + [delta][mu], it is necessary that t should exceed the
value given by the following equation:--
t = [lambda]/[delta][mu] (7).
8. _Diffraction Gratings._--Under the heading "Colours of Striated
Surfaces," Thomas Young (_Phil. Trans._, 1802) in his usual summary
fashion gave a general explanation of these colours, including the law
of sines, the striations being supposed to be straight, parallel and
equidistant. Later, in his article "Chromatics" in the supplement to the
5th edition of this encyclopaedia, he shows that the colours "lose the
mixed character of periodical colours, and resemble much more the
ordinary prismatic spectrum, with intervals completely dark interposed,"
and explains it by the consideration that any phase-difference which may
arise at neighbouring striae is multiplied in proportion to the total
number of striae.
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).
[Illustration: FIG. 6.]
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.
[Illustration: FIG. 7.]
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].
[Illustration: FIG. 8.]
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.
[Illustration:
| | | | | ( ( ( | | | | ) | (
FIG. 9.--x^2. FIG. 10.--y^2. FIG. 11.--x^3. FIG. 12.--xy^2.
/ / /
\ | | / | \ | | | |
/ / /