v - [Lambda]dv/d[Lambda] (2).
The signs of dv and d[Lambda] being opposite, the abnormal
displacement is in addition to the normal effect of the prism. But,
since dv/d[Lambda], or dv/d[lambda], is not constant, the achromatism
of the white band is less perfect than when no prism is used.
If a grating were substituted for the prism, v would vary as [Lambda],
and (2) would vanish, so that in all orders of spectra the white band
would be seen undisplaced.
In optical experiments two trains of waves can interfere only when
they have their origin in the same source. Otherwise, as it is usually
put, there can be no permanent phase-relation, and therefore no
regular interference. It should be understood, however, that this is
only because trains of optical waves are never absolutely homogeneous.
A really homogeneous train could maintain a permanent phase-relation
with another such train, and, it may be added, would of necessity be
polarized in its character. The peculiarities of polarized light with
respect to interference are treated under POLARIZATION OF LIGHT.
In a classical experiment interference-bands were employed to examine
whether light moved faster or slower in glass than in air. For this
purpose a very thin piece of glass may be interposed in the path of
one of the interfering rays, and the resulting displacement of the
bands is such as to indicate that the light passing through the glass
is _retarded_. In a better form of the experiment two pieces of
parallel glass cut from the same plate are interposed between the
prism and the screen, so that the rays from O1 (fig. 1) pass through
one part and those from O2 through the other. So long as these pieces
are parallel, no shifting takes place, but if one be slightly turned,
the bands are at once displaced. In the absence of dispersion the
retardation R due to the plate would be independent of [lambda], and
therefore completely compensated at the point determined by u = DR/b;
but when there is dispersion it is accompanied by a fictitious
displacement of the fringes on the principle explained by Airy, as was
shown by Stokes.
Before quitting this subject it is proper to remark that Fresnel's
bands are more influenced by diffraction than their discoverer
supposed. On this account the fringes are often unequally broad and
undergo fluctuations of brightness. A more precise calculation has
been given by H. F. Weber and by H. Struve, but the matter is too
complicated to be further considered here. The observations of Struve
appear to agree well with the corrected theory.
S 8. _Colours of Thin Plates._--These colours, familiarly known as those
of the soap-bubble, are seen under a variety of conditions and were
studied with some success by Robert Hooke under the name of "fantastical
colours" (_Micrographia_, 1664). The inquiry was resumed by Sir Isaac
Newton with his accustomed power ("Discourse on Light and Colours,"
1675, _Opticks_, book ii.), and by him most of the laws regulating these
phenomena were discovered. Newton experimented especially with thin
plates of air enclosed by slightly curved glasses, and the coloured
rings so exhibited are usually called after him "Newton's rings."
The colours are manifested in the greatest purity when the reflecting
surfaces are limited to those which bound the thin film. This is the
case of the soap-bubble. When, as is in other respects more
convenient, two glass plates enclosing a film of air are substituted,
the light under examination is liable to be contaminated by that
reflected from the outer surfaces. A remedy may be found in the use of
wedge-shaped glasses so applied that the outer surfaces, though
parallel to one another, are inclined to the inner operating surfaces.
By suitable optical arrangements the two portions of light, desired
and undesired, may then be separated.
In his first essay upon this subject Thomas Young was able to trace
the formation of these colours as due to the interference of light
reflected from the two surfaces of the plate; or, as it would be
preferable to say, to the superposition of the two reflected
vibrations giving resultants of variable magnitude according to the
phase-relation. A difficulty here presents itself which might have
proved insurmountable to a less acute inquirer. The luminous vibration
reflected at the second surface travels a distance increased by twice
the thickness of the plate, and it might naturally be supposed that
the relative retardation would be measured by this quantity. If this
were so, the two vibrations reflected from the surfaces of an
infinitely thin plate would be in accordance, and the intensity of the
resultant a maximum. The facts were notoriously the reverse. At the
place of contact of Newton's glasses, or at the thinnest part of a
soap-film just before it bursts, the colour is black and not white as
the explanation seems to require. Young saw that the reconciliation
lies in the circumstance that the two reflections occur under
different conditions, one, for example, as the light passes from air
to water, and the second as it passes from water to air. According to
mechanical principles the second reflection involves a change of sign,
equivalent to a gain or loss of half an undulation. When a series of
waves constituting any particular coloured light is reflected from an
infinitely thin plate, the two partial reflections are in absolute
discordance and, if of equal intensity, must give on superposition
complete darkness. With the aid of this principle the sequence of
colours in Newton's rings is explained in much the same way as that of
interference fringes (above, S 5).
[Illustration: FIG. 2.]
The complete theory of the colours of thin plates requires us to take
account not merely of the two reflections already mentioned but of an
infinite series of such reflections. This was first effected by S. D.
Poisson for the case of retardations which are exact multiples of the
half wave-length, and afterwards more generally by Sir G. B. Airy
(_Camb. Phil. Trans._, 1832, 4, p. 409).
In fig. 2, ABF is the ray, perpendicular to the wave-front, reflected
at the upper surface, ABCDE the ray transmitted at B, reflected at C
and transmitted at D; and these are accompanied by other rays
reflected internally 3, 5, &c., times. The first step is to calculate
the retardation [delta] between the first and second waves, so far as
it depends on the distances travelled in the plate (of index [mu]) and
in air.
If the angle ABF = 2[alpha], angle BCD = 2[alpha]' and the thickness
of plate = t, we have
[delta] = [mu](BC + CD) - BG
= 2[mu]BC - 2BC sin [alpha] sin [alpha]' = 2[mu]BC(l - sin^2 [alpha]')
= 2[mu]t cos [alpha]' (1).
In (1) [alpha]' is the angle of refraction, and we see that, contrary
to what might at first have been expected, the retardation is least
when the obliquity is greatest, and reaches a maximum when the
obliquity is zero or the incidence normal. If we represent all the
vibrations by complex quantities, from which finally the imaginary
parts are rejected, the retardation [delta] may be expressed by the
introduction of the factor [epsilon]^(-i[kappa][delta]), where i =
[root](-1), and [kappa] = 2[pi]/[lambda].
At each reflection or refraction the amplitude of the incident wave
must be supposed to be altered by a certain factor which allows room
for the reversal postulated by Young. When the light proceeds from the
surrounding medium to the plate, the factor for reflection will be
supposed to be b, and for refraction c; the corresponding quantities
when the progress is from the plate to the surrounding medium will be
denoted by e, f. Denoting the incident vibration by unity, we have
then for the first component of the reflected wave b, for the second
cef[Epsilon]^(-i[kappa][delta]), for the third
ce^3f[epsilon]^{-2i[kappa delta]}, and so on. Adding these together,
and summing the geometric series, we find
cef[epsilon]^(-i[kappa][delta])
b + ----------------------------------- (2).
1 - e^2[epsilon]^(-i[kappa][delta])