Stoke's Test

Around 1867, Stokes did a series of experiments to determine which of the two theories was most nearly conformable to facts.

He found the construction of Huygens and Fresnel to be decidedly the more correct. The difference between the results of it and the rival construction being about 100 times the probable error of observation.[58]

The hypothesis that in crystals the inertia depends on direction seemed therefore to be discredited when the theory based on it was compared with the results of observation.

But when, in 1888, W. Thomson (Lord Kelvin) revived Cauchy’s. theory of the labile aether, the question naturally arose as to whether that theory could be extended so as to account for the optical properties of crystals.

R. T. Glazebrook[59] showed that the correct formulae of crystal-optics are obtained when the Cauchy-Thomson hypothesis of zero velocity for the longitudinal wave is combined with the Stokes-Rankine-Rayleigh hypothesis of aelotropic inertia.

For on reference to the formulae which have been already given, it is obvious that the equation of motion of an aether having these properties must be

where e denotes the displacement, n the rigidity, and (ρ1, ρ2, ρ3) the inertia: and this equation leads by the usual analysis to Fresnel’s wave-surface.

The displacement e of the aethereal particles is not, however, accurately in the wave-front, as in Fresnel’s theory, but is at right angles to the direction of the ray, in the plane passing through the ray and the wave-normal.[60]

Having now traced the progress of the elastic-solid theory so far as it is concerned with the propagation of light in ordinary isotropic media and in crystals, we must consider the attempts which were made about this time to account for the optical properties of a more peculiar class of substances.

In 1811[61], Arago found that the state of polarization of a beam of light is altered when the beam is passed through a plate of quartz along the optic axis.

The phenomenon was studied shortly afterwards by Biot,[62] who showed that the alteration consists in a rotation of the plane of polarization about the direction of propagation: the angle of rotation is proportional to the thickness of the plate and inversely proportional to the square of the wave-length.

In some specimens of quartz the rotation is from left to right, in others from right to left. This distinction was shown by Sir John Herschel[63] (b. 1792, d. 1871) in 1820 to be associated with differences in the crystalline forn of the specimens, the two types bearing the same relation to each other as a right-handed and left-handed helix respectively.

Fresnel[64] and W. Thomson[65] proposed the term helical to denoto the property of rotating the plane of polarization, exhibited by such bodies as quartz: the less appropriato term natural rotatory polarization is, however, generally used.[66]

Biot showed that many liquid organic bodies, e.g. turpentine and sugar solutions, possess the natural rotatory property: we might be led to infer the presence of a helical structure in the molecules of such substances.

This inference is supported by the study of their chemical constitution; for they are invariably of the mirror-image’ or “enantiomorphous” type, in which one of the atoms (generally carbon) is asymmetrically linked to other atoms.

The next advance in the subject was due to Fresnel,[67] who showed that in naturally active bodies the velocity of propagation of circularly polarized light is different according as the polarization is right-handed or left-handed.

From this property the rotation of the plane of polarization of a plane. polarized ray may be immediately deduced; for the plane-polarized ray may be resolved into two rays circularly polarized in opposite senses, and these advance in phase by different amounts in passing through a given thickness of the substance: at any stage they may be recompounded into a plane-polarized ray, the azimuth of whose plane of polarization varies with the length of path traversed.

A ray of light incident on a crystal of quartz will in general bifurcate into two refracted rays, each of which will be elliptically polarized, i.e. will be capable of resolution into two plane-polarized components which differ in phase by a definite amount.

The directions of these refracted rays may be determined by Huygens’ construction, provided the wave-surface is supposed to consist of a sphere and spheroid which do not touch.

The first attempt to frame a theory of naturally active bodies was made by MacCullagh in 1836.[68] Suppose a plane wave of light to be propagated within a crystal of quartz. Let (x, y, z) denote the coordinates of a vibrating molecule, when the axis of x is taken at right angles to the plane of the wave, and the axis of z at right angles to the axis of the crystal.

Using Y and Z to denote the displacements parallel to the axes of y and z respectively at any time t, MacCullagh assumed that the differential equations which determine Y and Z are

where μ denotes a constant on which the natural rotatory property of the crystal clepends. In order to avoid complications arising from the ordinary crystalline properties of quartz, we shall suppose that the light is propagated parallel to the optic axis, so that we can take c1 equal to c2.

Assuming first that the beam is circularly polarized, let it be represented by

the ambiguous sign being determined according as the circular polarization as right-handed or left-handed. Substituting in the above differential equations, we have

or

Since 1/l denotes the velocity of propagation, it is evident that the reciprocals of the velocities of propagation of a right-handed and left-handed beam differ by the quantity

from which it is easily shown that the angle through which the plane of polarization of a plane-polarized beam rotates in unit length of path is

If we neglect the variation of c, with the period of the light, this expression satisfies Biot’s law that the angle of rotation in unit length of path is proportional to the inverse square of the wave-length.

MacCullagh’s investigation can be scarcely called a theory, for it amounts only to a reduction of the phenomena to empirical, though mathematical, laws; but it was on this foundation that later workers built the theory which is now accepted. [69]

The great investigators who developed the theory of light after the death of Fresnel devoted considerable attention to the optical properties of metals. Their researches in this direction must now be reviewed.