The values of the wave velocities for different directions can be best appreciated by constructing the normal surface. This is a two-sheeted surface constructed by laying off in every direction from an origin two radii propor tional to the two wave velocities. Its general form can be seen from fig. 18 which shows a perspec tive drawing of a portion of it, which is repeated by reflection in the principal planes. The figure is drawn on the assumption that a>0>7, and it will be seen that the two sheets meet in four conical points one in each quadrant of the plane of xz, that is in the plane perpendicular to the intermediate axis. Waves going in the directions of these conical points will have the same velocity whatever their polarization, and so for these directions light will not be polarized. These are the two optic axes which give rise to the name biaxial. Uniaxial crystals may be regarded as a degenerate case in which the two optic axes have approached one another; the normal surface becomes a sphere and an oval surface, which touch along the direction of the axis. The normal surface does not show how the waves corresponding to the two sheets are polarized, and for this Fresnel gave a very convenient construction. The ovaloid is an oval surface obtained by laying off a radius in each direction 1: m: n according to the rule When this surface is cut by a plane parallel to the wave front, the longest and shortest radii of the section give the two values of the wave-velocity, and their directions give the polarizations. This construction shows among other things that the wave-velocity is fixed when the polarization is given, without reference to the direction of the wave-front.
The phenomena we have so far described suffice to explain many of the features of crystal optics, in particular they are all that is required to understand the action of quarter-wave plates, nicols and other polarizing instruments, but they do not explain the fundamental fact that things seen through a crystal look double. To understand this we have to consider rays, not plane waves of indefinite breadth. In making Huygens' construction it would be wrong to draw the normal surface round each point and base the construction of fig. 3 on this, for the normal surface is only a diagram describing how plane waves can go, and does not represent the front of a wave emitted from a point. To find the form of this wave-front, we imagine that at every point of the normal surface a plane is drawn perpendicular to the radius vector. All these planes will envelop a surface of two sheets, and this we call the ray surface. In general shape it resembles the normal surface, and has four conical points lying in the same plane, but now at different angles; these are called the ray axes. Huygens' construction is done with the ray surface, not the nor mal surface. For uniaxial crystals the ray surface degenerates to a sphere and a spheroid touching the sphere at the ends of the axis.
It will be readily believed that double refraction involves much complicated geometry, and the complete conquest of the sub ject by Fresnel is one of the greatest feats ever performed in physics. Effects can be obtained by illuminating crystals with
suitably polarized light. Plate fig. 4 shows the effect obtained in the case of a uniaxial crystal, cut at right angles to the crystal axis; and fig. 5 that in the case of a biaxial crystal, cut at right angles to the bisector of the angle between the optic axes. We must omit their explanation, which requires a detailed discussion. We can only refer to the curious phenomenon of conical refrac tion which was discovered theoretically by Hamilton and after wards verified. When a narrow beam is sent along the axis of a biaxial crystal, the direction for the ray becomes indeterminate so that it can be anywhere on a certain cone. On emergence at the other side this cone is made into a cylinder by the surface refraction, and if this falls on a screen we get a ring of light.
Double refraction is invariably present in crystals which are not of the regular system, but is often quite small. Even in a strongly doubly refracting crystal like calcite the two principal indices are 1.66 and 1.49 so that their difference is considerably less than the refractive effect of either (which may be represented by its difference from unity). In uniaxial crystals and in biaxial of the orthorhombic system the axes are fixed by the crystal sym metry, though the principal wave-velocities may vary with the colour. In biaxial crystals of the monoclinic and triclinic systems the principal axes may vary in position as well, and the most com plicated colour pattern. may be produced. Some crystals, such as tourmaline, show a selective absorption, so that one of the two polarized waves cannot penetrate far into the crystal, and the light emerges from the other side plane polarized.
Double refraction also occurs when an isotropic solid is in a state of strain, and indeed the chance strains in badly annealed glass are sometimes a cause of trouble in experiments with polar ized light. On the other hand advantage has been taken of the effect, for by making a transparent model, say of a girder, it is possible to find the strains set up in it by the appropriate forces in cases where the shape is too complicated for direct calculation. Another occurrence of double refraction is the Kerr effect,—an ambiguous name, as there is a second effect of magnetic type, also named after this investigator. When light is sent through the glass of a charged electric condenser, double refraction occurs, so that the component polarized in the direction of the electric force has wave-velocity slightly different from that transversely polarized; the effect is proportional to the square of the electric force across the condenser. Yet another case, predicted and discovered by Voigt is a very small double refraction when light traverses matter placed in a strong magnetic field at right angles; this is associated with magnetic gyration which we shall discuss later.