PO will represent the ordinary wave-front in the crystal and PE the extraordinary wave-front, and the line Qo will represent the direction of the ordinary ray and QE the direction of the extraordinary ray. If the plane of the paper had not been a principal plane, we should have had to draw through P a plane perpendicular to the plane of incidence to touch the spheroid, and it would not have touched it at a point in the plane of the paper; so that the extraordinary ray would not be in the plane of incidence, and thus would not have obeyed the first law of refraction as given in $341. Two particular cases are worth examining: first, when the optic axis is parallel to the face of the crystal and perpendicular to the plane of incidence; and second, when the optic axis is parallel to the face and also parallel to the plane of incidence. In the first case (Fig. 388) the optic axis is perpendicular to the plane of the paper, and hence the sections of the wave-surfaces consist of two in the crystal being Hence in Fig. 388, the velocity of the extraordinary ray in a plane at right angles to the optic axis, the radius of the spherical wave-surface taken as a, the major axis of the spheroid will be b. if qo is a, QE will be equal to b. If the velocity of light in air is c, the refractive index for the ordinary ray is c/a, and that for the ordinary ray in a plane at right angles to the optic axis is cb. Now b or QE is constant for all angles of incidence in a plane at right angles to the optic axis, and hence the extraordinary refractive index is constant in this plane, and the extraordinary ray obeys the ordinary laws of refraction. By cutting a prism of Iceland spar with its refracting edge parallel to the optic axis two refracted rays will be obtained, and the refractive index (c/a and cb) corresponding to each of these can be measured. In this way it can be proved that the extraordinary refractive index (c/b) in a plane at right angles to the optic axis is constant. Hence b or QE must be constant, and so it is proved that the section of the extraordinary wave-surface perpendicular to the axis is a circle. X AXIS FIG. 389. M The construction for finding the directions of the refracted rays when the optic axis is parallel to the face of the crystal, and in the plane of incidence, is shown in Fig. 389. 406. Nicol's Prism.-As a means of obtaining plane polarised light, a tourmaline plate is, for many purposes, unsuited, for, as has been mentioned, the light transmitted by tourmaline is coloured green. Since, when a beam of light is passed through The most convenient method of getting rid of one of the rays is to make use of total internal reflection for this purpose. A rhomb of Iceland spar is taken and cut in two by a plane, AC (Fig. 390), perpendicular to the principal plane for the face AB. ૦ The two surfaces are then polished and cemented together in their original position by means of a thin film of Canada balsam. Now the refractive index of Canada balsam (1.55) is greater than the minimum value for the extraordinary ray (1.486) in Iceland spar, and less than that for the ordinary ray (1.658). As total reflection can only occur when light is passing from a media of greater to one of less refractive index, we can never get total reflection in the case of the extraordinary ray when passing from spar to balsam, so long as the ray passes in such a direction that the refractive index is less than 1.55. In the case of the ordinary ray, however, if the incidence is sufficiently oblique we shall obtain total reflection. Hence if the plane AC is suitably inclined, the ordinary ray, PO, will be incident on the surface AC at an angle greater than the critical angle, and will therefore be totally reflected along oo', while the extraordinary ray, PEE', will pass through the prism. The light transmitted by such a rhomb of Iceland spar, which is called a Nicol's prism, will therefore be plane polarised, and since it is the extraordinary ray which is transmitted, the plane of polarisation is perpendicular to the principal plane, i.e. is a plane perpendicular to the paper in Fig. 390. A Nicol's prism may be used, not only for producing plane polarised light, when it is called a polariser, but also for detecting whether light is plane polarised, and, if so, determine the plane in which it is polarised, when it is said to be used as an analyser. If the light incident on the Nicol is unpolarised, then the intensity of the transmitted light will remain the same when the Nicol is rotated round the light ray as an axis, the intensity of the transmitted light being practically half that of the incident light. There is, however, a very slight loss due to reflection at E (Fig. 390), and where the ray leaves the crystal. If the incident light is plane polarised, the intensity of the transmitted light varies as the analyser is rotated. When the principal plane of the Nicol is parallel to the plane of polarisation of the incident ray, then (401) there will be only an ordinary ray in the spar, and this ray is totally reflected, so that no light will be transmitted. When the principal plane of the Nicol is perpendicular to the plane of polarisation of the incident light, only an extraordinary ray will be produced in the spar, and this will be transmitted undiminished, so that in this case the intensity of the transmitted light is equal to that of the incident light. If the principal plane of the Nicol is inclined at an angle a to the plane of polarisation, it can be shown, exactly as in § 401, that the intensity of the extraordinary ray, and hence that of the transmitted light, is I sin a, where I is the intensity of the incident light. Thus when a=0 or 180° the intensity of the transmitted light is zero, and when a = 90° or 270° the intensity of the transmitted light is I. 407. Polarisation by Reflection.-If the light reflected from a non-metallic surface, such as glass, is examined with an analysing Nicol, it will be found that as the Nicol is rotated the intensity of the transmitted light varies. For a certain angle of incidence there is no light transmitted by the Nicol when its principal plane is parallel to the plane of incidence of the reflected light, while when the principal plane of the Nicol is perpendicular to the plane of incidence, the transmitted light is equal in intensity to the reflected light before it passes through the Nicol. This shows that, for this angle of incidence, the reflected ray is completely plane polarised in the plane of incidence. For all other angles of incidence the reflected ray is only partly polarised, i.e. consists of a mixture of ordinary unpolarised light with light which is polarised in the plane of incidence. The angle of incidence, for which the reflected beam is completely plane polarised, is called the polarising angle for the reflecting substance. If, instead of consisting of ordinary light, the incident ray is plane polarised, and is incident at the polarising angle, then when the incident ray is polarised in the plane of incidence, i.e. the vibrations of the ether are taking place perpendicular to the plane of incidence, the light will be reflected. If, however, the incident ray is polarised in a plane perpendicular to the plane of incidence, so that the ether vibrations take place in this plane, none of the light will be reflected, but it will all be refracted into the reflecting substance. If the incident light is polarised in intermediate planes, the reflected light will gradually increase in intensity as the plane of polarisation changes from the position in which it is perpendicular to the plane of incidence, to that in which it is parallel to the plane of incidence. Owing to polarisation by reflection, a glass plate can be used both as a polariser and as an analyser. N' Ν' Suppose a ray of light 10 (Fig. 391) is incident on a plate of glass A, at the polarising angle, which for ordinary glass is about 56°. The reflected ray, OP, will be polarised in the plane of incidence, that is, in the plane of the paper. If this reflected ray is received on a second glass mirror, B, also at the polarising angle, then if the plane of incidence on B is parallel to the plane of incidence on A, as is shown at (a) and (b), the N. polarised light will be reflected along PR. If now the mirror B is rotated round an axis parallel to OP, the angle of incidence will remain the same, viz. equal to the polarising angle, but the intensity of the reflected N A (a) (b) FIG. 391. ray will diminish until, when the plane of incidence, which is of course the plane passing through P and 0, and containing the normal PN', is perpendicular to the plane of the paper, there will be no reflected ray. Of course the reflected ray will again be of maximum intensity when the mirror has been turned through 180° into the position B', and zero when it has been turned through 270°. Thus the one glass plate, A, has acted as a polariser and the other, B, as an analyser. This is the principle of Biot's and of Norrenberg's polariscopes. 408. Brewster's Law.-Sir David Brewster, having made an extensive series of experiments on the angle of polarisation for different substances, found that the tangent of the angle of polarisation is equal to the refractive index of the substance,1 or if is the angle of polarisation and μ the refractive index N N' FIG. 392. P R -B. μ=tan . The geometrical interpretation of Brewster's law is very interesting. Let 10 (Fig. 392) be a ray of light incident on a reflecting surface at the polarising angle 4, and OR and OP be the direction of the reflected and refracted rays respectively. If ẞ is the angle of refraction Hence the angle POB, which is equal to 90°-B, is equal to 4, or to the angle NOR. Adding the angle ROB to each, we get that the angle POR is equal to the angle BON. Hence, since the angle BON is a right angle, the angle POR must also be a right angle; that is, when the angle 1 More recent observations by Jamin have shown that Brewster's law is only exact for substances for which μ is about 1.46. For substances of refracted index differing much from this value, the reflected beam is never entirely plane polarised, but for an angle of incidence given by the relation tan =μ the quantity of plane polarised light is a maximum. |