We must also consider the nature of ordinary unpolarized light. This has no "sides" and is symmetrical about the direction of propagation, and in this it is like circularly polarized light ; but we shall see that circularly polarized light can easily be converted into plane polarized, whereas for ordinary light this is not possible, so the explanation of its symmetry must be found in another way. If the light is rigorously monochromatic, the phases of the corn ponents df the light-vector are maintained for ever, and such light must therefore be polarized, either plane, circularly or ellip tically. In nature, however, no light is rigorously monochromatic, for the Michelson interferometer fails to show interference for path differences of more than a million wave-lengths, and this enables us to explain unpolarized light as being polarized light of which the direction is changed frequently. We suppose each atom to emit a polarized monochromatic wave which lasts for a time corresponding to a million wave-lengths, and then changes and emits one of a different polarization. The length of time of each separate type is so short that we cannot distinguish them, but merely get an average of all the different types and this aver age will be symmetrical about the direction of propagation.
In describing plane polarized light we have said that it is polar ized in the direction x when the light-vector vibrates along x. Un happily there is some confusion in the terminology used. When polarization was first discovered the convention was adopted that the light of the ordinary ray in double refraction is polarized in the plane through the axis of the crystal and the direction of propagation. It later appeared that the light-vector must be per pendicular to this plane, so that the old convention would have described the light as polarized in the plane of yz, which we call polarized in the direction x; and it was even sometimes loosely referred to the direction y alone. For most optical effects the be haviour is determined by the direction of the vector and not by the direction of propagation of the wave, so that it is practically inconvenient to mention two directions when one would do. The modern tendency, which we shall adopt, is to describe plane polar ized light by the direction of the light-vector ; thus, in double re fraction we shall say that the ordinary ray is polarized at right angles to the crystal axis. Some confusion is unavoidable because the term plane polarized (instead of line polarized) is too deeply implanted to be changed, but we can avoid ambiguity by saying that light is plane polarized in some direction, instead of in some plane, and the direction is then the normal to the plane which was formerly used.
Double refraction is not the only way in which polarization is produced, and we shall discuss its other occurrences in detail later. Chief among them is reflection at a transparent surface set at a suitable angle. At other angles there is an incomplete polarization, which is sometimes very troublesome in experiments ; for as light passes through any system of lenses or mirrors, it is almost im possible to prevent its becoming polarized, and this may easily ob scure the study of its original polarization. We may also mention
the polarization of scattered light, exemplified in the light of the sky. Again, in the diffraction of light by a slit, the two compo-, nents behave rather differently, and so the light is slightly polar ized ; the theory we have given is sufficiently correct for small angles and we shall not consider the modification that is rigorously required. Perhaps the most interesting occurrence of all is in the Zeeman effect (q.v.). Here the spectral lines of an atom become displaced and split when the atom is in a strong magnetic field, and the split components are polarized in certain definite ways, from which much information can be obtained about the mechan ics of the atom.
The Analysis of Polarized Light.—Polarization is chiefly, though by no means exclusively, studied by using the property of double refractinn in crystals. We discuss this below, but it will here suffice to say that when a wave goes through a crystal its light-vector is resolved in two directions, mutually perpendicular and perpendicular to the direction of propagation, and the com ponents in these directions have different wave-velocities. Suppose that we have a plate with parallel faces, and that the wave is going through it along z, while the two special directions are x and y. In such a plate the light-vector will be given by where a and b are the wave-velocities for the directions x and y. If the incident light is plane polarized along x or y the wave will go through unchanged, but if it is polarized, say at angle y on entering the plate at z= o, we shall have a = j3 and A= Ccosy, B= Csiny. If the plate is of thickness 1 the light will emerge from the farther side as and so X and Y are no longer in the same phase, and the light is now elliptically polarized. Fig. 15 shows diagrammatically how this transformation comes about. If we had started with ellip tically polarized light, the emergent light would in general be elliptically polarized, but in a new direction. In special cases it might become plane polarized, and this explains the principle of an important instrument used in the study of polarized light, the quarter-wave plate. A quarter-wave plate is made by taking 1 so that 27r-pl/ a— 2.nwl/b= go°. If the incident light is elliptically polarized, with the axes of the ellipse along x and y, we may write it as After passing through the quarter-wave plate the emergent light will be and it is now plane polarized. As a case of special importance, if the incident light is circularly polarized, the emergent will be plane polarized at 45 degrees to the x direction, and its right- or lef t handedness will be indicated according to whether the direction is between x and +y or x and —y. Quarter-wave plates are usually made of mica, which is easily cleft into thin sheets. These sheets exhibit a rather weak double refraction, so that a quarter-wave plate is not inconveniently thin. As mica is rather soft, it is often mounted between two sheets of glass and scratches on these indi cate the direction of the crystal axis.