LIGHT 613
Fig. 35. components are the same, or differ by an integral multiple of >.
3. It becomes a circle when the amplitudes of the components are equal, and their phases differ by an odd multiple of i tr. The motion takes place in one direction (say right-handedly) in the circle when this multiplier is 1, 5, 9, 13, &c., and in the opposite (left-handed) when it is 3, 7, 11, 15, &c. . .
Effect of a plate of doubly refracting material.
Homogeneous light.
Now, suppose a plane polarized ray to fall on a plate of a doubly-refracting crystal (a thin plate of mica orselenite, for instance). Within the plate it will in general be divided into two, which are polarized in planes at right material, angles to one another. The directions of vibration in these rays are determined by the physical properties of the material. Let them be represented by the lines Ox, Oy in fig. 35. Then, if OA represents the semiampli- tude of vibration in the incident ray, it may be looked on by (2) above as the resultant of two simple harmonic motions of the same period, whose semiamplitudes are OM and OX, and which are in the same phase. Each of these will pass through the plate of crystal unchanged. But one will, in general, travel faster than the other; for the essential cause of double refraction is the difference of velocities of the two rays. The portions of the two rays which simultaneously escape from the crystal, and which travel together outside it, will therefore differ in phase. Hence, to find the nature of the transmitted light, we must recombine the vibrations in OM, OX, taking account of this dif ference of phase. By (1) above the result will be in general elliptic motion. The ellipse will necessarily be one of the infinite number which can be inscribed in the rectangle AA BB , whose construction is obvious. We have then, in general, what is called elliptically polarized light. This degenerates (by (2) above) into plane polarized light, whose vibrations are along OA or OA according as the Difference O f phase is 0, 2-rr, kir, <tc., or tr, 3-rr, 5-n-, &c. And it will become circularly polarized light if OM = OX (i.e., if A0.r = ^7r) and the difference of phase be an odd
multiple of ir. By (3) above this will be right or left
handed, according to the value of the odd multiplier.
Elliptically and circuarly polarized light.
This conclusion from the assumption above made is fully borne out by experiment. When a plate of mica, of such a thickness as to retard one of the two rays a quarter of a wave-length more than the other, is interposed between two Nicols, we observe the following phenomena: –
If the Nicols were originally placed so as to extinguish the light, the introduction of the mica plate in general partially restores it. Now, let the mica plate be made to rotate in its own plane. The light vanishes for successive positions, differing by a quadrant of rotation, i.e. whenever the directions of vibration in the crystal coincide with the principal planes of the Nicols. In each of these positions the light from the first Nicol passes unchanged through the mica, and is therefore entirely stopped by the second Nicol. Half-way between these positions the light trans mitted through the system is at its brightest; and in these cases it is not altered in brightness by rotating the second Nicol. It is then circularly polarized, and in whatever direction the second Nicol is placed the component of the circular motion which is ready to pass through it is of the same amplitude. Here, then, is a case in which a Nicol (the second) cannot enable us to distinguish between com mon light and light very seriously modified.
White light.
In what precedes, we have assumed that homogeneous light was used. In general, a doubly-refracting plate produces a difference of phase in its two rays which will depend on their wave-length; and thus when white light is used we have a display of colour, sometimes extremely gorgeous, and we may distinguish light thus circularly polarized from common light by slight changes of colour and intensity as the second Nicol is turned.
Effect of total reflexion.
Fresnel's rhomb.
Hitherto we have spoken of the polarizing angle for light reflected in air from bodies such as glass, water, &c., which have a higher refractive index than air, and we have seen that an equal amount of light is polarized in the refracted beam. But what if there be no refracted beam? This is the case of total reflexion inside the denser body. Fresnel discovered that in this case the two kinds of polarized light (in planes at right angles to one another) co-exist in the totally reflected ray, but that they differ in phase, and therefore in general recombine into elliptically polarized light. Guided by peculiar theoretical considerations, he was led to construct a piece of glass (Fresnel's rhomb), inside which light is twice totally reflected at a certain angle with the result that, if it be originally polarized in a plane inclined at 45 to the plane of reflexion, the emergent light is circularly polarized.
Metallic reflexion.
Reflexion from the surface of metals, and of very highly refractive substances such as diamond, generally gives at all incidences elliptically polarized light. Attempts have been made to determine from such effects the refractive indices of metals and other opaque substances. These are all based upon theory, and cannot as yet command much confidence. With certain doubly-refracting substances the light reflected at a definite angle is differently polarized, and sometimes even differently coloured, for different azimuths of the plane of incidence.
Rings and cross in plate of uniaxal crystal.
When a thin plate of doubly-refracting crystal, which gives a bright colour when placed between two Nicols, is slightly inclined to the ray, the colour changes as the difference of phase of the two refracted rays is increased. If, now, we take a plate of Iceland spar cut perpendicularly to the axis, no colour will be produced by parallel rays passing through it perpendicularly, because both rays have a common velocity parallel to the axis; but, if divergent light be used, there is a gorgeous display of circular coloured rings surrounding the axis, which depends upon the increasing retardation of the ordinary ray behind the extraordinary as their inclination to the axis increases. When the principal planes of the Nicols are at right angles, this system of rings is intersected by two black diameters, in these planes respectively. When the second Nicol is turned through a right angle, we have exactly the complement of the former appearance, i.e., a figure such that, if superposed on the former, it would give an uniform field of white light.
It is to be noticed that none of these phenomena can be observed without the use of the second Nicol. This arises from the fact that, where the vibrations in any direction interfere so as to destroy one another, those in the direction perpendicular to the former interfere so as to strengthen one another. The second Nicol enables us to select one of these portions, and examine it independently of the other.
Biaxal crystals.
The only double refraction we have considered particu- larly is that of Iceland spar, where everything is symme- trical about the axis of the crystal. Such crystals, and they include as a rule all those of the second and third systems in CRYSTALLOGRAPHY (q.v.), are called uniaxal. Crystals of the first system are not doubly refractive. But it was one of the most valuable of Brewster's discoveries that the great majority of non-isotropic substances are doubly refracting, and in general are biaxal, i.e., have two
