R = 2t[theta]^2/[mu] (1).
as sufficiently approximate.
The condition of distinctness is here satisfied, since R is the same
for every ray emergent parallel to a given one. The rays of one
parallel system are collected by the lens to a focus at a definite
point in the neighbourhood of the original source.
The formula (1) was discussed by Herschel, and shown to agree with
Newton's measures. The law of formation of the rings follows
immediately from the expression for the retardation, the radius of the
ring of n^(th) order being proportional to n and to the square root of
the wave-length.
S 10. _Interferometer._--In many cases it is necessary that the two rays
ultimately brought to interference should be sufficiently separated over
a part of their course to undergo a different treatment; for example, it
may be desired to pass them through different gases.
[Illustration: FIG. 6.]
A simple modification of Young's original experiment suffices to solve
this problem. Light proceeding from a slit at A (fig. 6) perpendicular
to the plane of the paper, falls upon a collimating lens B whose
aperture is limited by two parallel and rather narrow slits of equal
width. The parallel rays CE, DF (shown broken in the figure)
transmitted by these slits are brought to a focus at G by the lens EF
where they form an image of the original slit A. This image is
examined with an eye-piece of high magnifying power. The interference
bands at G undergo displacement if the rays CE, DF are subjected to a
relative retardation. Consider what happens at the point G, which is
the geometrical image of A. If all is symmetrical so that the paths
CE, DF are equal, there is brightness. But if, for example, CE be
subjected to a relative retardation of half a wave-length, the
brightness is replaced by darkness, and the bands are shifted through
half a band-interval.
[Illustration: FIG. 7.]
An apparatus of this kind has been found suitable for determining the
refractivity of gases, especially of gases available only in small
quantities (_Proc. Roy. Soc._, 1896, 59, p. 198; 1898, 64, p. 95).
There is great advantage in replacing the ordinary eye-piece by a
simple cylindrical magnifier formed of a glass rod 4 mm. in diameter.
Under these conditions a paraffin lamp sufficed to illuminate the slit
at A, and allowed the refractivities of gases to be compared to about
one-thousandth part.
If the object be to merely see the bands in full development the
lenses of the above apparatus may be dispensed with. A metal or
pasteboard tube 10 in. long carries at one end a single slit
(analogous to A) and at the other a double slit (analogous to C, D).
This double slit, which requires to be very fine, may be made by
scraping two parallel lines with a knife on a piece of silvered glass.
The tube is pointed to a bright light, and the eye, held close behind
the double slit, is focused upon the far slit.
S 11. _Other Refractometers._--In another form of refractometer,
employed by J. C. Jamin, the separations are effected by reflections
at the surfaces of thick plates. Two thick glass mirrors, exactly the
same in all respects, are arranged as in fig. 7. The first of the two
interfering rays is that which is reflected at the first surface of
the first reflector and at the second surface of the second reflector.
The second ray undergoes reflection at the second surface of the first
reflector and at the first surface of the second reflector. Upon the
supposition that the plates are parallel and equally thick, the paths
pursued by these two rays are equal. P represents a thin plate of
glass interposed in the path of one ray, by which the bands are
shifted.
[Illustration: FIG. 8.]
In Jamin's apparatus the two rays which produce interference are
separated by a distance proportional to the thickness of the mirrors,
and since there is a practical limit to this thickness, it is not
possible to separate the two rays very far. In A. A. Michelson's
interferometer there is no such restriction. "The light starts from
source S (fig. 8) and separates at the rear of plate A, part of it
being reflected to the plane mirror C, returning exactly, on its path
through A, to O, where it may be observed by a telescope or received
upon a screen. The other part of the ray goes through the glass plate
A, passes through B, and is reflected by the plane mirror D, returns
on its path to the starting point A, where it is reflected so as
nearly to coincide with the first ray. The plane parallel glass B is
introduced to compensate for the extra thickness of glass which the
first ray has traversed in passing twice through the plate A. Without
it the two paths would not be optically identical, because the first
would contain more glass than the second. Some light is reflected from
the front surface of the plate A, but its effect may be rendered
insignificant by covering the rear surface of A with a coating of
silver of such thickness that about equal portions of the incident
light are reflected and transmitted. The plane parallel plates A and B
are worked originally in one piece, which is afterwards cut in two.
The two pieces are placed parallel to one another, thus ensuring exact
equality in the two optical paths AC and AD" (see Michelson,
_Light-Waves and their Uses_, Chicago, 1903).
The adjustments of this apparatus are very delicate. Of the fully
silvered mirrors C, D, the latter must be accurately parallel to the
image of the former. For many purposes one of the mirrors, C, must be
capable of movement parallel to itself, usually requiring the use of
very truly constructed ways. An escape from this difficulty may be
found in the employment of a layer of mercury, standing on copper, the
surface of which automatically assumes the horizontal position.
Michelson's apparatus, employed to view an extended field of
homogeneous light, exhibits Haidinger's rings, and if all is in good
order the dark parts are sensibly black. As the order of interference
increases, greater and greater demand is made upon the homogeneity of
the light. Thus, if the illumination be from a sodium flame, the rings
are at first distinct, but as the difference of path increases the
duplicity of the bright sodium line begins to produce complications.
After 500 rings, the bright parts of one system coincide with the dark
parts of the other (Fizeau), and if the two systems were equally
bright all trace of rings would disappear. A little later the rings
would again manifest themselves and, after 1000 had gone by, would be
nearly or quite as distinct as at first. And these alternations of
distinctness and indistinctness would persist until the point was
reached at which even a single sodium line was insufficiently
homogeneous. Conversely, the changes of _visibility_ of the rings as
the difference of path increases give evidence as to the duplicity of
the line. In this way Michelson obtained important information as to
the constitution of the approximately homogeneous lines obtained from
electrical discharge through attenuated metallic vapours. Especially
valuable is the vacuum tube containing cadmium. The red line proved
itself to be single and narrow in a high degree, and the green line
was not far behind.
But although in Michelson's hands the apparatus has done excellent
spectroscopic work, it is not without its weak points. A good deal of
labour is required to interpret the visibility curves, and in some
cases the indications are actually ambiguous. For instance, it is
usually impossible to tell on which side of the principal component a
feebler companion lies. It would seem that for spectroscopic purposes
this apparatus must yield to that of Fabry and Perot, in which
multiple reflections are utilized; this is a spectroscope in the
literal sense, inasmuch as the constitution of a spectrum line is seen
by simple inspection. (R.)