since the sines are practically equal to the angles, as are also the tangents. But we know that sin i/sin r=n, and when the angles are small, i/r n. It is thus easy to show that the coefficient of reflection for normal incidence It is interesting to find out how much light is reflected at normal incidence from the surface of water, for example For water, n is 4/3, so that the reflection coefficient is(1/7)2 s 2%. At normal incidence, only two percent of the light is reflected from the surface of water 33-7 Anomalous refraction The last polarization effect we shall consider was actually one of the first to be discovered anomalous refraction. Sailors visiting Iceland brought back to Europe crystals of Iceland spar(CaCO3)which had the amusing property of mak ing anything seen through the crystal appear doubled, i. e, as two images. This came to the attention of Huygens, and played an important role in the discovery of polarization. As is often the case, the phenomena which are discovered first are the hardest, ultimately, to explain. It is only after we understand a physical concept thoroughly that we can carefully select those phenomena which most clearly and simply demonstrate the concept Anomalous refraction is a particular case of the same birefringence that w considered earlier. Anomalous refraction comes about when the optic axis, the long axis of our asymmetric molecules, is not parallel to the surface of the crystal In Fig. 33-7 are drawn two pieces of birefringent material, with the optic axis as shown. In the upper figure, the incident beam falling on the material is linearly polarized in a direction perpendicular to the optic axis of the material. When this beam strikes the surface of the material, each point on the surface acts as a source aB of a wave which travels into the crystal with velocity v, the velocity of light in the crystal when the plane of polarization is normal to the optic axis. The wave front is just the envelope or locus of all these little spherical waves, and this wave- front moves straight through the crystal and out the other side This is just the dinary behavior we would expect, and this ray is called the ordinary ray In the lower figure the linearly polarized light falling on the crystal has its direction of polarization turned through 90, so that the optic axis lies in the plane of polarization. When we now consider the little waves originating at any point on the surface of the crystal, we see that they do not spread out as spherical waves tion is perpendicular to the optic axis, whereas the light travelling perpendicular the path of the ordinary rar 0i Light travelling along the optic axis travels with velocity u, because the polariza- Fig. 33-7. The upper diagram to the optic axis travels with velocity vi because the polarization is parallel to the doubly refracting crystal.The extraor complete analysis will show that the waves spread out on the surface of an ellipsoid, gram. The optic axis lies in the plane of with the optic axis as major axis of the ellipsoid. The envelope of all these elliptical the paper. waves is the wavefront which proceeds through the crystal in the direction shown Again, at the back surface the beam will be deflected just as it was at the front surface, so that the light emerges parallel to the incident beam, but displaced from it. Clearly, this beam does not follow Snell,'s law, but goes in an extraordinary direction. It is therefore called the extraordinary ray When an unpolarized beam strikes an anomalously refracting crystal, it is separated into an ordinary ray, which travels straight through in the normal man ner, and an extraordinary ray which is displaced as it passes through the crysta These two emergent rays are linearly polarized at right angles to each other. That his is true can be readily demonstrated with a sheet of polaroid to analyze the polarization of the emergent rays. We can also demonstrate that our interpretation of this phenomenon is correct by sending linearly polarized light into the crystal By properly orienting the direction of polarization of the incident beam make this light go straight through without splitting, or we can make it go through 33-9