consideration increases with ε; 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/λ. 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 12x = P; or since x is small,
In like manner, from observations upon another radiation λ′ to be compared with λ, we have
whence if t be treated as an absolute constant,
| λ′ | = | P | ( 1 + | x2 | − | x′2 | ) |
| λ | P′ | 8 | 8 |
The ratio λ/λ′ 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 λ/λ′ 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 λ′/λ.
In order to find P we may employ a modified form of (16), viz.,
| P′ | = | λ | ( 1 + | x2 | − | x′2 | ) |
| P | λ′ | 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 Pérot’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).
§ 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.
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| 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 μ, 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 θ, 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 2μt + BI, and of the latter from B to the equivalent place G is 2μBF. Now FB = t sec θ′, θ′ being the angle of refraction; BI = 2t tan θ′sin θ; so that the relative retardation F is given by
If θ, θ′ be small, we may take
| R = 2tθ2/μ | (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 nth order being proportional to n and to the square root of the wave-length.
§ 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.
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| Fig. 6. |
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| Fig. 7. |
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.
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.
§ 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
