[lambda] sin [alpha]' d[alpha]' + cos [alpha]' d[lambda] = 0,
where ([alpha] being constant)
cos [alpha]' d[alpha]' = sin [alpha] d[mu].
[lambda] d[mu]
Thus cot^2 [alpha]' = - -------- --------- (9),
[mu] d[lambda]
giving [alpha]' when the relation between [mu] and [lambda] is known.
According to A. L. Cauchy's formula, which represents the facts very
well throughout most of the visible spectrum,
[mu] = A + B[lambda]^(-2) (10),
so that
2B 2([mu] - A)
cot^2 [alpha]' = -------------- = ----------- (11).
[lambda]^2[mu] [mu]
If we take, as for Chance's "extra-dense flint," B = .984 X 10^(-10),
and as for the soda lines, [mu] = 1.65, [lambda] = 5.89 X 10^(-6), we
get
[alpha]' = 79 deg. 30'.
At this angle of refraction, and with this kind of glass, the
retardation of phase is accordingly nearly independent of wave-length,
and therefore the bands formed, as the thickness varies, are
approximately achromatic. Perfect achromatism would be possible only
under a law of dispersion
[mu]^2 = A' - B'[lambda]^2.
If the source of light be distant and very small, the black bands are
wonderfully fine and numerous. The experiment is best made (after
Newton) with a right-angled prism, whose hypothenusal surface may be
brought into approximate contact with a plate of black glass. The
bands should be observed with a convex lens, of about 8 in. focus. If
the eye be at twice this distance from the prism, and the lens be held
midway between, the advantages are combined of a large field and of
maximum distinctness.
If Newton's rings are examined through a prism, some very remarkable
phenomena are exhibited, described in his twenty-fourth observation
(_Opticks_; see also Place, _Pogg. Ann._, 1861, 114, 504). "When the
two object-glasses are laid upon one another, so as to make the rings
of the colours appear, though with my naked eye I could not discern
above eight or nine of those rings, yet by viewing them through a
prism I could see a far greater multitude, insomuch that I could
number more than forty.... And I believe that the experiment may be
improved to the discovery of far greater numbers.... But it was on but
one side of these rings, namely, that towards which the refraction was
made, which by the refraction was rendered distinct, and the other
side became more confused than when viewed with the naked eye....
"I have sometimes so laid one object-glass upon the other that to the
naked eye they have all over seemed uniformly white, without the least
appearance of any of the coloured rings; and yet by viewing them
through a prism great multitudes of those rings have discovered
themselves."
Newton was evidently much struck with these "so odd circumstances";
and he explains the occurrence of the rings at unusual thicknesses as
due to the dispersing power of the prism. The blue system being more
refracted than the red, it is possible under certain conditions that
the n^(th) blue ring may be so much displaced relatively to the
corresponding red ring as _at one part of the circumference_ to
compensate for the different diameters. A white stripe may thus be
formed in a situation where without the prism the mixture of colours
would be complete, so far as could be judged by the eye.
The simplest case that can be considered is when the "thin plate" is
bounded by plane surfaces inclined to one another at a small angle. By
drawing back the prism (whose edge is parallel to the intersection of
the above-mentioned planes) it will always be possible so to adjust
the effective dispersing power as to bring the n^(th) bars to
coincidence for any two assigned colours, and therefore approximately
for the entire spectrum. The formation of the achromatic band, or
rather central black band, depends indeed upon the same principles as
the fictitious shifting of the centre of a system of Fresnel's bands
when viewed through a prism.
But neither Newton nor, as would appear, any of his successors has
explained why the bands should be more numerous than usual, and under
certain conditions sensibly achromatic for a large number of
alternations. It is evident that, in the particular case of the
wedge-shaped plate above specified, such a result would not occur. The
width of the bands for any colour would be proportional to [lambda],
as well after the displacement by the prism as before; and the
succession of colours formed in white light and the number of
perceptible bands would be much as usual.
The peculiarity to be explained appears to depend upon the _curvature_
of the surfaces bounding the plate. For simplicity suppose that the
lower surface is plane (y = 0), and that the approximate equation of
the upper surface is y = [alpha] + bx^2, a being thus the least
distance between the plates. The black of the n^(th) order for
wave-length [lambda] occurs when
(1/2)n[lambda] = [alpha] + bx^2 (12);
and thus the width ([delta]x) at this place of the band is given by
(1/2)[lambda] = 2bx[delta]x (13);
[lambda] [lambda]
or [delta]x = -------- = ----------------------------------------- (14).
4bx 4[root]b.[root]((1/2)n[lambda] - [alpha])
If the glasses be in contact, as is usually supposed in the theory of
Newton's rings, [alpha] = 0, and [delta]x[oo][lambda]^(1/2), or the
width of the band of the n^(th) order varies as the square root of the
wave-length, instead of as the first power. Even in this case the
overlapping and subsequent obliteration of the bands is greatly
retarded by the use of the prism, but the full development of the
phenomenon requires that [alpha] should be finite. Let us inquire what
is the condition in order that the width of the band of the n^(th)
order may be stationary, as [lambda] varies. By (14) it is necessary
that the variation of [lambda]^2/((1/2)n[lambda] - [alpha]) should
vanish. Hence a = (1/4)n[lambda], so that the interval between the
surfaces at the place where the n^(th) band is formed should be half
due to curvature and half to imperfect contact at the place of closest
approach. If this condition be satisfied, the achromatism of the
n^(th) band, effected by the prism, carries with it the achromatism of
a large number of neighbouring bands, and thus gives rise to the
remarkable effects described by Newton. Further developments are
given by Lord Rayleigh in a paper "On Achromatic Interference Bands"
(_Phil. Mag._, 1889, 28, pp. 77, 189); see also E. Mascart, _Traite
d'optique_.
In Newton's rings the variable element is the thickness of the plate,
to which the retardation is directly proportional, and in the ideal
case the angle of incidence is constant. To observe them the eye is
focused upon the thin plate itself, and if the plate is very thin no
particular precautions are necessary. As the plate thickens and the
order of interference increases, there is more and more demand for
homogeneity in the light, and we may have recourse to a sodium-flame
or a helium vacuum tube. At the same time the disturbing influence of
obliquity increases. Unless the aperture of the eye is reduced, the
rays reaching it from even the same point of the plate are differently
affected, and complications ensue tending to impair the distinctness
of the bands. To obviate this disturbance it is best to work at
incidences as nearly as possible perpendicular.
[Illustration: FIG. 4.]
The bands seen when light from a soda flame falls upon nearly parallel
surfaces are often employed as a test of flatness. Two flat surfaces
can be made to fit, and then the bands are few and broad, if not
entirely absent; and, however the surfaces may be presented to one
another, the bands should be straight, parallel and equidistant. If
this condition be violated, one or other of the surfaces deviates from
flatness. In fig. 4, A and B represent the glasses to be tested, and C
is a lens of 2 or 3 ft. focal length. Rays diverging from a soda flame
at E are rendered parallel by the lens, and after reflection from the
surfaces are recombined by the lens at E. To make an observation, the
coincidence of the radiant point and its image must be somewhat
disturbed, the one being displaced to a position a little beyond, and
the other to a position a little in front of the diagram. The eye,
protected from the flame by a suitable screen, is placed at the image,
and being focused upon AB, sees the field traversed by bands. The
reflector D is introduced as a matter of convenience to make the line
of vision horizontal.
These bands may be photographed. The lens of the camera takes the
place of the eye, and should be as close to the flame as possible.
With suitable plates, sensitized by cyanin, the exposure required may
vary from ten minutes to an hour. To get the best results, the hinder
surface of A should be blackened, and the front surface of B should be
thrown out of action by the superposition of a wedge-shaped plate of
glass, the intervening space being filled with oil of turpentine or
other fluid having nearly the same refraction as glass. Moreover, the
light should be purified from blue rays by a trough containing
solution of bichromate of potash. With these precautions the dark
parts of the bands are very black, and the exposure may be prolonged
much beyond what would otherwise be admissible.
By this method it is easy to compare one flat with another, and thus,
if the first be known to be free from error, to determine the errors
of the second. But how are we to obtain and verify a standard? The
plan usually followed is to bring _three_ surfaces into comparison.
The fact that two surfaces can be made to fit another in all azimuths
proves that they are spherical and of equal curvatures, but one convex
and the other concave, the case of perfect flatness not being
excluded. If A and B fit one another, and also A and C, it follows
that B and C must be similar. Hence, if B and C also fit one another,
all three surfaces must be flat. By an extension of this process the
errors of three surfaces which are not flat can be found from a
consideration of the interference bands which they present when
combined in three pairs.
The free surface of undisturbed water is almost ideally flat, and, as
Lord Rayleigh (_Nature_, 1893, 48, 212) has shown, there is no great
difficulty in using it as a standard of comparison. Following the same
idea we may construct a parallel plate by superposing a layer of water
upon mercury. If desired, the superior reflecting power of the mercury
may be compensated by the addition of colouring matter to the water.
_Haidinger's Rings dependent on Obliquity._--It is remarkable that the
well-known theoretical investigation, undertaken with the view of
explaining Newton's rings, applies more directly to a different system
of rings discovered at a later date.
The results embodied in equations (1) to (8) have application in the
first instance to plates whose surfaces are absolutely parallel,
though doubtless they may be employed with fair accuracy when the
thickness varies but slowly.
We have now to consider t constant and [alpha]' variable in (1). If
[alpha]' be small,
[delta] = 2[mu]t(1 - (1/2)[alpha]'^2) = 2[mu]t - t[alpha]^2/[mu] (15);
and since the differences of [delta] are proportional to [alpha]^2,
the law of formation is the same as for Newton's rings, where [alpha]'
is constant and t proportional to the square of the distance from the
point of contact. In order to see these rings distinctly the eye must
be focused, not upon the plate, but for infinitely distant objects.
The earliest observation of rings dependent upon obliquity appears to
have been made by W. von Haidinger (_Pogg. Ann._, 1849, 77, p. 219;
1855, 96, p. 453), who employed sodium light reflected from a plate of
mica (e.g. 0.2 mm. thick). The transmitted rays are the easier to see in
their completeness, though they are necessarily somewhat faint. For this
purpose it is sufficient to look through the mica, held close to the eye
and perpendicular to the line of vision, at a sheet of white paper or
card illuminated by a sodium flame. Although Haidinger omitted to
consider the double refraction of the mica and gave formulae not quite
correct for even singly refracting plates, he fully appreciated the
distinctive character of the rings, contrasting _Beruhrungsringe und
Plattenringe_. The latter may appropriately be named after him. Their
tardy discovery may be attributed to the technical difficulty of
obtaining sufficiently parallel plates, unless it be by the use of mica
or by the device of pouring water upon mercury. Haidinger's rings were
rediscovered by O. R. Lummer (_Wied. Ann._, 1884, 23, p. 49), who
pointed out the advantages they offer in the examination of plates
intended to be parallel.
The illumination depends upon the intensity of the monochromatic
source of light, and upon the reflecting power of the surfaces. If R
be the intensity of the reflected light we have from (7)
1 (1 - e^2)^2
--- = 1 + --------------------------------;
R 4e^2 sin^2 ((1/2)[kappa][delta])
from which we see that if e = 1 absolutely, 1/R = R = 1 for all values
of [delta]. If e = 1 very nearly, R = 1 nearly for all values of
[delta] for which sin^2((1/2)[kappa][delta]) is not very small. In the
light reflected from an extended source, the ground will be of full
brightness corresponding to the source, but it will be traversed by
_narrow_ dark lines. By transmitted light the ground, corresponding to
general values of the obliquity, will be dark, but will be interrupted
by narrow bright rings, whose position is determined by sin
(1/2)([kappa][delta]) = 0. In permitting for certain directions a
complete transmission in spite of a high reflecting power (e) of the
surfaces, the plate acts the part of a resonator.
There is no transparent material for which, unless at high obliquity,
e approaches unity. In C. Fabry and A. Perot's apparatus the
reflections at nearly perpendicular incidence are enhanced by lightly
silvering the surfaces. In this way the advantage of narrowing the
bright rings is attained in great measure without too heavy a
sacrifice of light. The plate in the optical sense is one of air, and
is bounded by plates of glass whose inner silvered surfaces are
accurately flat and parallel. The outer surfaces need only ordinary
flatness, and it is best that they be not quite parallel to the inner
ones. The arrangement constitutes a _spectroscope_, inasmuch as it
allows the structure of a complex spectrum line to be directly
observed. If, for example, we look at a sodium flame, we see in
general two distinct systems of narrow bright circles corresponding to
the two D-lines. With particular values of the thickness of the plate
of air the two systems may coincide so as to be seen as a single
system, but a slight alteration of thickness will cause a separation.
It will be seen that in this apparatus the optical parts are
themselves of extreme simplicity; but they require accuracy of
construction and adjustment, and the demand in these respects is the
more severe the further the ideal is pursued of narrowing the rings by
increase of reflecting power. Two forms of mounting are employed. In
one instrument, called the _interferometer_, the distance between the
surfaces--the thickness of the plate--is adjustable over a wide range.
In its complete development this instrument is elaborate and costly.
The actual measurements of wave-lengths by Fabry and Perot were for
the most part effected by another form of instrument called an
_etalon_ or interference-gauge. The thickness of the optical plate is
here fixed; the glasses are held up to metal knobs, acting as
distance-pieces, by adjustable springs, and the final adjustment to
parallelism is effected by regulating the pressure exerted by these
springs. The distance between the surfaces may be 5 or 10 mm.
The theory of the comparison of wave-lengths by means of this
apparatus is very simple, and it may be well to give it, following
closely the statement of Fabry and Perot (_Ann. chim. phys._, 1902,
25, p. 110). Consider first the cadmium radiation [lambda] treated as
a standard. It gives a system of rings. Let P be the ordinal number of
one of these rings, for example the first counting from the centre.
This integer is supposed known. The order of interference at the
centre will be p = P + [epsilon]. We have to determine this number
[epsilon], lying ordinarily between 0 and 1. The diameter of the ring
under consideration increases with [epsilon]; so that a measure of
the diameter allows us to determine the latter. Let t be the thickness
of the plate of air. The order of interference at the centre is p =
2t/[lambda]. This corresponds to normal passage. At an obliquity i the
order of interference is p cos i. Thus if x be the angular diameter of
the ring P, p cos (1/2)x = P; or since x is small,
p = P(1 + (1/8)x^2).
In like manner, from observations upon another radiation [lambda]' to
be compared with [lambda], we have
p' = P'(1 + (1/8)x'^2);
whence if t be treated as an absolute constant,
[lambda]' P / x^2 x'^2 \
--------- = --- ( 1 + --- - ---- ) (16).
[lambda] P' \ 8 8 /
The ratio [lambda]/[lambda]' is thus determined as a function of the
angular diameters x, x' and of the integers P, P'. If P, say for the
cadmium red line, is known, an approximate value of [lambda]/[lambda]'
will usually suffice to determine what integral value must be assigned
to P', and thence by (16) to allow of the calculation of the corrected
ratio [lambda]'/[lambda].
In order to find P we may employ a modified form of (16), viz.,
P' [lambda] / x^2 x'^2 \
--- = --------- ( 1 + --- - ---- ) (17).
P [lambda]' \ 8 8 /
using spectrum lines, such as the cadmium red and the cadmium green,
for which the relative wave-lengths are already known with accuracy
from A. A. Michelson's work. To test a proposed integral value of P
(cadmium red), we calculate P' (cadmium green) from (17), using the
observed values of x, x'. If the result deviates from an integer by
more than a small amount (depending upon the accuracy of the
observations), the proposed value of P is to be rejected. In this way
by a process of exclusion the true value is ultimately arrived at
(Rayleigh, _Phil. Mag._, 1906, 685). It appears that by Fabry and
Perot's method comparisons of wave-lengths may be made accurate to
about one-millionth part; but it is necessary to take account of the
circumstance that the effective thickness t of the plate is not
exactly the same for various wave-lengths as assumed in (16).
S 9. _Newton's Diffusion Rings._--In the fourth part of the second book
of his _Opticks_ Newton investigates another series of rings, usually
(though not very appropriately) known as the colours of thick plates.
The fundamental experiment is as follows. At the centre of curvature of
a concave looking-glass, quicksilvered behind, is placed an opaque card,
perforated by a small hole through which sunlight is admitted. The main
body of the light returns through the aperture; but a series of
concentric rings are seen upon the card, the formation of which was
proved by Newton to require the co-operation of the two surfaces of the
mirror. Thus the diameters of the rings depend upon the thickness of the
glass, and none are formed when the glass is replaced by a metallic
speculum. The brilliancy of the rings depends upon imperfect polish of
the anterior surface of the glass, and may be augmented by a coat of
diluted milk, a device used by Michel Ferdinand, duc de Chaulnes. The
rings may also be well observed without a screen in the manner
recommended by Stokes. For this purpose all that is required is to place
a _small_ flame at the centre of curvature of the prepared glass, so as
to coincide with its image. The rings are then seen surrounding the
flame and occupying a definite position in space.
The explanation of the rings, suggested by Young, and developed by
Herschel, refers them to interference between one portion of light
scattered or diffracted by a particle of dust, and then regularly
refracted and reflected, and another portion first regularly refracted
and reflected and then diffracted at emergence by the same particle.
It has been shown by Stokes (_Camb. Trans._, 1851, 9, p. 147) that no
regular interference is to be expected between portions of light
diffracted by different particles of dust.
In the memoir of Stokes will be found a very complete discussion of
the whole subject, and to this the reader must be referred who desires
a fuller knowledge. Our limits will not allow us to do more than touch
upon one or two points. The condition of fixity of the rings when
observed in air, and of distinctness when a screen is used, is that
the systems due to all parts of the diffusing surface should coincide;
and it is fulfilled only when, as in Newton's experiments, the source
and screen are in the plane passing through the centre of curvature of
the glass.
[Illustration: FIG. 5.]
As the simplest for actual calculation, we will consider a little
further the case where the glass is plane and parallel, of thickness t
and index [mu], and is supplemented by a lens at whose focus the
source of light is placed. This lens acts both as collimator and as
object-glass, so that the combination of lens and plane mirror
replaces the concave mirror of Newton's experiment. The retardation is
calculated in the same way as for thin plates. In fig. 5 the
diffracting particle is situated at B, and we have to find the
relative retardation of the two rays which emerge finally at
inclination [theta], the one diffracted at emergence following the
path ABDBIE, and the other diffracted at entrance and following the
path ABFGH. The retardation of the former from B to I is 2[mu]t + BI,
and of the latter from B to the equivalent place G is 2[mu]BF. Now FB
= t sec [theta]', [theta]' being the angle of refraction; BI = 2t tan
[theta]'sin[theta]; so that the relative retardation F is given by
R = 2[mu]t{1 + [mu]^(-1) tan [theta]' sin [theta] -sec [theta]'}
= 2[mu]t(1 - cos [theta]').