[=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]
[Illustration: FIG. 17.]
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.