in the crystalline arrangement of the molecules, as it is lost on fusion or solution, and in several cases belongs to enantiomorphous crystals, the two correlated forms of which are the one right-handed and the other left-handed optically as well as crystallographically, this being necessarily the case if the property be retained when the crystal is fused or dissolved. In organic bodies the rotary property, as the researches of J. A. Le Bel, J. H. van’t Hoff and others have established, corresponds to the presence of one or more asymmetric atoms of carbon—that is, atoms directly united to elements or radicles all different from one another—and in every case there exists an isomer that rotates the plane of polarization to the same degree in the opposite direction. Absence of rotary power when asymmetric carbon atoms are present, may be caused by an internal compensation within the molecule as with the inactive tartaric acid (meso-tartaric acid), or may be due to the fact that the compound is an equimolecular mixture of left- and right-hand varieties, this being the case with racemic acid that was broken by Louis Pasteur into laevo- and dextro-tartaric acid (see Stereo-Isomerism).
Substances that by reason of the structure or arrangement of their molecules rotate the plane of polarization are said to be structurally active, and the rotation produced by unit length is called their rotary power. If unit mass of a solution contain m grammes of an active substance and if δ be the density and ρ be the rotary power of the solution, the specific rotary power is defined by ρ/mδ, and the molecular rotary power is obtained from this by multiplying by the hundredth part of the molecular mass. This quantity is not absolutely constant, and in many cases varies with the concentration of the solution and with the nature of the solvent. A mixture of two active substances, or even of an active and an inactive substance, in one solution sometimes produces anomalous effects.
Fresnel showed that rotary polarization could be explained kinematically by supposing that a plane-polarized stream is resolved on entering an active medium into two oppositely circularly polarized streams propagated with different speeds, the rotation being right- or left-handed according as the right- or left-handed stream travels at the greater rate.
The polarization-vector of the primitive stream being ξ＝a cos nt, the first circularly polarized stream after traversing a distance z in the medium may be represented by
ξ1＝a cos (nt − k1z), η1＝a sin (nt − k1z),
and the second by
ξ2＝a cos (nt − k2z), η2＝ − a sin (nt − k2z).
The resultant of these is
ξ＝2a cos 12 (k2 − k1)z cos {nt − 12(k2 + k2)z},
η＝2a sin 12 (k2 − k1)z cos {nt − 12(k1 + k2)z},
which shows that for any fixed value of z the light is plane polarized in a plane making an angle 12(k2−k1)z＝π(λ2−1−λ1−1)z, with the initial plane of polarization, λ1 and λ2 being the wave-lengths of the circular components of the same frequency.
Since the two circular streams have different speeds, Fresnel ar ued that it would be possible to separate them by oblique refraction, and though the divergence is small, since the difference of their refractive indices in the case of quartz is only about 0·00007, he succeeded by a suitable arrangement of alternately right- and left-handed prisms of quartz in resolving a plane-polarized stream into two distinct circularly polarized streams. A similar arrangement was used by Ernst v. Fleischl for demonstrating circular polarization in liquids. This result is not, however, conclusive; for an application of Huygens’s principle shows that it is a consequence of the rotation of the plane of polarization by an amount proportional to the distance traversed, independently of the state of affairs within the active medium. Not more convincing is a second experiment devised by Fresnel. If in the interference experiment with Fresnel's mirrors or biprism the slit be illuminated with white light that has passed through a polarize and a quartz plate cut perpendicularly to the optic axis. it is found on analysing the light that in addition to the ordinary central set of coloured fringes two lateral systems are seen, one on either side of it. According to Fresnel’s explanation the light in each of the interfering streams consists of to trains of waves that are circularly polarized in opposite direction and have a relative retardation of phase, introduced by the passage through the quartz the central fringes are then due to the similarly polarized waves; the lateral systems are produced by the oppositely polarized streams, these on analysation being capable of interfering A. Righi has, however, pointed out that this experiment may be explained by the fact that the function of the quartz plate and analyser is to eliminate the constituents of the composite stream of white light that mask the interference actually occurring at the positions of the lateral systems of fringes, and that any other method of removing them is equally effective. In fact, the lateral systems are obtained when a plate of selenite is substituted for the quartz.
Sir G. B. Airy extended Fresnel’s hypothesis to directions inclined to the axis of uniaxal crystals by assuming that in any such direction the two waves, that can be propagated without alteration of their state of polarization, are oppositely elliptically polarized with their planes of maximum polarization parallel and perpendicular to the principal plane of the wave, these becoming practically plane polarized at a small inclination to the optic axis. Several investigations have been made to test the correctness of Airy’s views, but it must be remembered that it is only possible to experiment on waves after they have left the crystal, and) L. G. Gouy (Journ. de phys., 1885 2], iv. 149) has shown that the results deduced from Airy's waves of permanent type may be obtained by regarding the action of the medium as the superposition of the effects of ordinary double refraction and of an independent rotary power. As regards the course of the streams on refraction into the crystal, it is found that it is determined by the Huygenian law (see Refraction, § Double); as, however, the two streams in the direction of the axis have different speeds, the spherical and the spheroidal sheets of the wave surface do not touch as in the case of inactive uniaxal crystals. On these principles Airy, by an elaborate mathematical investigation, successfully explained the interference patterns obtained with plates of quartz perpendicular to the optic axis. When the polarize and analyser are parallel or crossed, the pattern is the same as with inactive plates, with the exception that the brushes do not extend to the centre of the field; but as the analyser is rotated a small cross begins to appear at the centre of the field, while the rings change their form and become nearly squares with rounded corners when the planes of polarization and analysation are at 45°. With two plates of equal thickness and of opposite rotations the pattern consists of a series of circles and of four similar spirals starting from the centre, each spiral being turned through 90° from that adjacent to it. When the light is circularly polarized or circularly analysed, a single plate gives two mutually inwrapping spirals, and similar spirals in circularly polarized light are obtained with plates of an active biaxal crystal perpendicular to one of the optic axes. It was in this way that the rotary property of certain biaxal crystals was first established by Pocklington.
F. E. Reusch has shown that a packet of identical inactive plates arranged in spiral fashion gives an artificial active system, and the behaviour of certain pseudo symmetric crystals indicates a formation of this character. On these results L. Sohncke (Math. Ann., 1876, ix. 504) and E. Mallard (Traité de cristallographie, vol. ii. ch. ix.) have built up a theory of the structure of active media, but in the instances in which static spirality has been shown to be effective in producing optical rotation the coarse-grainedness of the structure is comparable with the wave-length of the radiation affected.
The rotary property may be induced in substances naturally inactive. Thus A. W. Ewell (Amer. Jour. of Science, 1899 [4], viii. 89) has shown the existence of a rotational effect in twisted glass and gelatine, the rotation being opposite to the direction of the twist. But a far more important instance of induced activity is afforded by Michael Faraday’s discovery of the rotary polarization connected with a magnetic field. There is, however, a marked difference between this magnetic rotation and that of a structurally active medium, for in the latter it is always right-handed or always left-handed with respect to the direction of the ray, while in the former the sense of rotation is determined by the direction of magnetization and therefore remains the same though the ray be reversed. This subject is treated in the article Magneto-Optics, to which the reader is also referred for John Kerr’s discovery of the effect on polarization produced by reflection from a magnetic pole, and for the action of a magnetic field on the radiation of a source—the “Zeeman effect.”
Reflection and Refraction.—Huygens satisfactorily explained the laws of reflection and refraction on the principles of the wave theory, so far as the direction of the waves is concerned, but his explanation gives no account of the intensity and the polarization of the reflected light. This was supplied by Fresnel, who, starting from a mechanical hypothesis, showed by ingenious but not strictly dynamical reasoning that if the incident stream have unit amplitude, that of the reflected stream will be
− sin (i − r )/sin (i + r ) or tan (i − r )/tan (i + r ),
according as the incident light is polarized in or perpendicularly to the plane of incidence i, r, being the angles of incidence and refraction connected by the formula sin i＝μ sin r. At normal incidence the intensity of the reflected light, measured by the square of the amplitude, is {(μ−1)/(μ+1)}2 in both cases; but whereas in the former the intensity increases uniformly with i to the value unity for i＝90°, in the latter the intensity at first decreases as i increases, until it attains the value zero when i+r＝90°, or tan i＝μ—the polarizing angle of Brewster—and then increases until it becomes unity at grazing incidence. If the incident light be polarized in a plane, making an angle α with the plane of incidence, the stream may be resolved into two that are polarized in the principal azimuths, and these will be reflected in accordance with the above laws. Hence if β be the angle between the plane of incidence and that in which the reflected light is polarized
tan β＝−tan α cos (i + r )/cos (i − r ).
The expressions for the intensity of the refracted light may be obtained from those relating to the reflected light by the principle of energy. In order to avoid the question of the measurements of the intensity in different media, it is convenient to suppose that the refracted stream emerges into a medium similar to the first by a transition so gradual that no light is lost by reflection. The intensities of the incident, reflected and refracted streams are then measured in the same way, and we have merely to express that
