(1 - cos [kappa][delta])^2 + sin^2 [kappa][delta]
= e^2 --------------------------------------------------------
(1 - e^2cos [kappa][delta])^2 + e^4 sin^2 [kappa][delta]
4e^2sin^2 {(1/2)[kappa][delta]}
= -------------------------------- (7);
1 - 2e^2 cos[kappa][delta] + e^4
(1 - e^2)^2
Intensity of transmitted light = -------------------------------- (8),
1 - 2e^2cos [kappa][delta] + e^4
the sum of the two expressions being unity.
According to (7) not only does the reflected light vanish completely
when [delta] = 0, but also whenever (1/2)[kappa][delta] = n[pi], n
being an integer, that is, whenever [delta] = n[lambda]. When the
first and third mediums are the same, as we have here supposed, the
central spot in the system of Newton's ring is _black_, even though
the original light contain a mixture of all wave-lengths. If the light
reflected from a plate of any thickness be examined with a
spectroscope of sufficient resolving power, the spectrum will be
traversed by dark bands, of which the centre corresponds to those
wave-lengths which the plate is incompetent to reflect. It is obvious
that there is no limit to the fineness of the bands which may be thus
impressed upon a spectrum, whatever may be the character of the
original mixed light.
The relations between the factors b, c, e, f have been proved,
independently of the theory of thin plates, in a general manner by
Stokes, who called to his aid the general mechanical principle of
_reversibility_. If the motions constituting the reflected and
refracted rays to which an incident ray gives rise be supposed to be
reversed, they will reconstitute a reversed incident ray. This gives
one relation; and another is obtained from the consideration that
there is no ray in the second medium, such as would be generated by
the operation alone of either the reversed reflected or refracted
rays. Space does not allow of the reproduction of the argument at
length, but a few words may perhaps give the reader an idea of how the
conclusions are arrived at. The incident ray (IA) (fig. 3) being 1,
the reflected (AR) and refracted (AF) rays are denoted by b and c.
When b is reversed, it gives rise to a reflected ray b^2 along AI, and
a refracted ray bc along AG (say). When c is reversed, it gives rise
to cf along AI, and ce along AG. Hence bc + ce = 0, b^2 + cf = 1, which
agree with (4). It is here assumed that there is no change of phase in
the act of reflection or refraction, except such as can be represented
by a change of sign.
[Illustration: FIG. 3.]
When the third medium differs from the first, the theory of thin
plates is more complicated, and need not here be discussed. One
particular case, however, may be mentioned. When a thin transparent
film is backed by a perfect reflector, no colours should be visible,
all the light being ultimately reflected, whatever the wave-length may
be. The experiment may be tried with a thin layer of gelatin on a
polished silver plate. In other cases where a different result is
observed, the inference is that either the metal does not reflect
perfectly, or else that the material of which the film is composed is
not sufficiently transparent. Some apparent exceptions to the above
rule, exhibited by thin films of collodion resting upon silver
surfaces, have been described by R. W. Wood (_Physical Optics_, p.
143), who attributes the very curious effects observed to _frilling_
of the collodion film.
For study of the colours of thin plates there are no more interesting
subjects than the soap-film. For projection the films may be stretched
across vertical rings of iron wire coated with paraffin. In their
undisturbed condition they thin from the top, and the colours are
disposed in horizontal bands. If, as suggested by Brewster, a jet of
wind issuing from a small nozzle and supplied from a well-regulated
bellows be allowed to impinge obliquely, parts of the film are set in
rotation, and displays of colours may be exhibited to a large
audience, astonishing by their brilliance and by the rapidity with
which they change. Permanent films, analogous to soap-films, are best
obtained by Glew's method. A few drops of celluloid varnish are poured
upon the surface of water contained in a large dish. After evaporation
of the solvent, the films may be picked up upon rings of iron wire.
As a variant upon Newton's rings, interesting effects may be obtained
by the partial etching of the surfaces of picked pieces of
plate-glass. A surface is coated in parallel stripes with paraffin wax
and treated with dilute hydrofluoric acid for such a time (found by
preliminary trials) as is required to eat away the exposed portions to
a depth of one quarter of the mean wave-length of light. Two such
prepared surfaces pressed in the crossed position into suitable
contact exhibit a chess-board pattern. Where two uncorroded, or where
two corroded, parts overlap, the colours are nearly the same; but
where a corroded and an uncorroded surface meet, a strongly contrasted
colour is developed. The combination lends itself to projection and
the pattern seen upon the screen is very beautiful if proper
precautions are taken to eliminate the white light reflected from the
first and fourth surfaces of the plates (see _Nature_, 1901, 64, 385).
Theory and observation alike show that the transmitted colours of a
thin plate, e.g. a soap film or a layer of air, are very inferior to
those reflected. Specimens of ancient glass, which have undergone
superficial decomposition, on the other hand, sometimes show
transmitted colours of remarkable brilliancy. The probable
explanation, suggested by Brewster, is that we have here to deal not
merely with one, but with a series of thin plates of not very
different thicknesses. It is evident that with such a series the
transmitted colours would be much purer, and the reflected much
brighter, than usual. If the thicknesses are strictly equal, certain
wave-lengths must still be absolutely missing in the reflected light;
while on the other hand a constancy of the interval between the plates
will in general lead to a special preponderance of light of some other
wave-length for which all the component parts as they ultimately
emerge are in agreement as to phase.
On the same principle are doubtless to be explained the colours of
fiery opals, and, more remarkable still, the iridescence of certain
crystals of potassium chlorate. Stokes showed that the reflected
light is often in a high degree monochromatic, and that it is
connected with the existence of twin planes. A closer discussion
appears to show that the twin planes must be repeated in a periodic
manner (_Phil. Mag._, 1888, 26, 241, 256; also see R. W. Wood, _Phil.
Mag._, 1906).
A beautiful example of a similar effect is presented by G. Lippmann's
coloured photographs. In this case the periodic structure is actually
the product of the action of light. The plate is exposed to stationary
waves, resulting from the incidence of light upon a reflecting surface
(see PHOTOGRAPHY).
All that can be expected from a physical theory is the determination
of the composition of the light reflected from or transmitted by a
thin plate in terms of the composition of the incident light. The
further question of the chromatic character of the mixtures thus
obtained belongs rather to physiological optics, and cannot be
answered without a complete knowledge of the chromatic relations of
the spectral colours themselves. Experiments upon this subject have
been made by various observers, and especially by J. Clerk Maxwell
(_Phil. Trans._, 1860), who has exhibited his results on a colour
diagram as used by Newton. A calculation of the colours of thin
plates, based upon Maxwell's data, and accompanied by a drawing
showing the curve representative of the entire series up to the fifth
order, has been given by Rayleigh (_Edin. Trans._, 1887). The colours
of Newton's scale are met with also in the light transmitted by a
somewhat thin plate of doubly-refracting material, such as mica, the
plane of analysis being perpendicular to that of primitive
polarization.
The same series of colours occur also in other optical experiments,
e.g. at the centre of the illuminated area when light issuing from a
point passes through a small round aperture in an otherwise opaque
screen.
The colours of which we have been speaking are those formed at nearly
perpendicular incidence, so that the retardation (reckoned as a
distance), viz. 2[mu]t cos [alpha]', as sensibly independent of [lambda].
This state of things may be greatly departed from when the thin plate
is rarer than its surroundings, and the incidence is such that
[alpha]' is nearly equal to 90 deg., for then, in consequence of the
powerful dispersion, cos [alpha]' may vary greatly as we pass from one
colour to another. Under these circumstances the series of colours
entirely alters its character, and the bands (corresponding to a
graduated thickness) may even lose their coloration, becoming sensibly
black and white through many alternations (Newton's _Opticks_, bk.
ii.; Fox-Talbot, _Phil. Mag._, 1836, 9, p. 40l). The general
explanation of this remarkable phenomenon was suggested by Newton.
Let us suppose that plane waves of white light travelling in glass are
incident at angle [alpha] upon a plate of air, which is bounded again
on the other side by glass. If [mu] be the index of the _glass_,
[alpha]' the angle of refraction, then sin [alpha]' = [mu] sin
[alpha]; and the retardation, expressed by the equivalent distance in
air, is
2t sec [alpha]' - [mu].2t tan [alpha]' sin [alpha] = 2t cos [alpha]';
and the retardation in _phase_ is 2t cos [alpha]'/[lambda], [lambda]
being as usual the wave-length in air.
The first thing to be noticed is that, when [alpha] approaches the
critical angle, cos[alpha]' becomes as small as we please, and that
consequently the retardation corresponding to a given thickness is
very much less than at perpendicular incidence. Hence the glass
surfaces need not be so close as usual.
A second feature is the increased brilliancy of the light. According
to (7) the intensity of the reflected light when at a maximum (sin
(1/2)[kappa][gamma] = 1) is 4e^2/(1 + e^2)^2. At perpendicular
incidence e is about 1/5, and the intensity is somewhat small; but, as
cos[alpha]' approaches zero, e approaches unity, and the brilliancy is
much increased.
But the peculiarity which most demands attention is the lessened
influence of a variation in [lambda] upon the phase-retardation. A
diminution of [lambda] of itself increases the retardation of phase,
but, since waves of shorter wave-length are more refrangible, this
effect may be more or less perfectly compensated by the greater
obliquity, and consequent diminution in the value of cos [alpha]'. We
will investigate the conditions under which the retardation of phase
is stationary in spite of a variation of [lambda].