Page:EB1911 - Volume 21.djvu/968

 exhibits in white light a distinctive greyish violet colour, known as a sensitive tint from the fact that it changes rapidly to blue or red, when the retardation is very slightly increased or diminished. If then the sensitive plate be cut in half and the two parts be placed side by side after the one has been turned through 90° in its own plane, the tint of the one half will be raised and that of the other will be lowered when the compound plate is associated with a second doubly refracting plate.

When light from an extended source is made to converge upon the crystal, the phenomenon of rings and brushes localized at infinity is obtained The exact calculation of the intensity in this case is very complicated and the resulting expression is too unwieldy to be of any use, but as an approximation the formula for the case of a parallel beam may be employed, the quantities and  therein occurring being regarded as functions of the angle and plane of incidence and consequently as variables. In monochromatic light, then, the interference pattern is characterized by three systems of curves. the curves of constant retardation = const; the lines of like polarization = const.; the curves of constant intensity  = const. When = 2n and also when  =  or +/2 or = or +/2, that is at points for which the streams within the plate are polarized in planes parallel and perpendicular to the planes of primitive and final polarization, the intensity (called the fundamental intensity) is the same as when the plate is removed. These conditions define to systems of curves called respectively the principal curves of constant retardation and the principal lines of like polarization, these latter lines dividing the field into regions in which the intensity is alternately greater and less than the fundamental intensity. When, however, the planes of polarization and analysation are parallel or crossed, the two pairs of principal lines of like polarization coincide, and the intensity is at all points in the former case not greater than, and in the latter case not less than, it was before the introduction of the plate. The determination of the curves of constant retardation depends upon expressing the retardation in terms of the optical constants of the crystal, the angle of incidence and the azimuth of the plane of incidence. P. A. Bertin has shown that a useful picture of the form of these curves may be obtained by taking sections, parallel to the plate, of a surface that he calls the “isochromatic surface,” and that is the locus of points on the crystal at which the relative retardation of two plane waves passing simultaneously through a given point and travelling in the same direction has an assigned value But as this surface is obtained by assuming that the interfering streams follow the same route in the crystal, and by neglecting the refraction out of the crystal, it does not lend itself to accurate numerical calculations. To the same degree of accuracy as that employed in obtaining the expression for the intensity, the form of the lines of like polarization is given by the section, parallel to the plate, of a cone, whose generating lines are the directions of propagation of waves that have their planes of polarization parallel and perpendicular to a given plane:the cone is in general of the third degree and passes through the optic axes of the cn stal We must limit ourselves in this article to indicating the chief features of the phenomenon in the more important cases. (Reference should be made to the article for illustrations, and for applications of these phenomena to the determination of crystal form.

With an uniaxial plate perpendicular to the optic axis, the curves of constant retardation are concentric circles and the lines of like polarization are the radii thus with polarize and analyser regulated for extinction, the pattern consists of a series of bright and dark circles interrupted by a black cross with its arms parallel to the planes of polarization and analysation. In the case of a biaxal plate perpendicular to the bisector of the acute angle between the optic axes, the curves of constant retardation are approximately Cassini’s ovals, and the lines of like polarization are equilateral hyperbola passing through the points corresponding to the optic axes. With a crossed polarizer and analyser the rings are interrupted by a dark hyperbolic brush that cuts the plane of the optic axes at right angles, if this plane be at 45° to the planes of polarization and analysation—the so-called diagonal position—and that becomes a rectan ular cross with its arms parallel and perpendicular to the plane of fine optic axes when this plane coincides with the plane of primitive or final polarization-the normal position.

When white light is employed coloured rings are obtained, provided the relative retardation of the interfering streams be not too great. The isochromatic lines, unless the dispersion be excessive, follow in the main the course of the curves of constant retardation and the principal lines of like polarization are with a crossed polarize and analyser dark brushes, that in certain cases are fringed with colour. This state of things may, however, be considerably departed from if the axes of optical symmetry of the crystal are different for the various colours. The examination of dispersion of the optic axes in biaxal crystals (see, § Double) may be conveniently made with a plate perpendicular to the acute bisectrix placed in the diagonal position for light of mean period between a crossed polarizer and analyser. When the rings are coloured symmetrically with respect to two perpendicular lines the acute bisectrix and the plane of the optic axes are the same for all frequencies, and the colour for which the separation of the axes is the least is that on the concave side of the summit of the hyperbolic brushes. Crossed, inclined and horizontal dispersion are characterized respectively by a distribution of colour that is symmetrical with respect to the centre alone, the plane of the optic axes, and the perpendicular plane.

The phenomenon of interference produced by crystalline plates is considerably modified if the light be circularly or elliptically polarized or analysed by the interposition of a quarter-wave between the crystal and the polarize or analyser. Thus in the two cases described above the brushes disappear and the rings are continuous when the light is both polarized and analysed circularly. But the most important case, on account of its practical application to determining the sign of a crystal, is that in which the light is plane polarized and circularly analysed or the reverse. Let us suppose that the light is circularly analysed and that the primitive and final planes of polarization are at right angles. Then with an uniaxal plate perpendicular to the optic axis, the black cross is replaced by two lines, on crossing which the rings are discontinuous, expansion or contraction occurring in the quadrants that contain the axis of the quarter-wave plate, according as the crystal is positive or negative. With a biaxal plate perpendicular to the optic axis in the diagonal position, the hyperbolic brush becomes an hyperbolic line and the rings are expanded or contracted on its concave side, with a positive plate, according as the plane of the optic axes is parallel or perpendicular to the axis of the quarter-wave plate, the reverse being the case with a negative plate.

With a combination of plates in plane-polarized and plane-analysed light the interference pattern with monochromatic light is generally very complicated, the dark curves when polarize and analyser are crossed being replaced b isolated dark spots or segments of lines. When, however, the field, is very small, or when the primitive light is white so that interference is only visible for small relative retardations, the problem becomes in many cases one of far less complexity. An instance of considerable importance is afforded by the combination known as Savart’s plate. This consists of two plates of an uniaxal crystal of equal thickness, cut at the same inclination of about 45° to the optic axis and superposed with their principal planes at right angles. The interference pattern produced by this combination is, when the field is small, a system of parallel straight lines bisecting the angle between the principal planes of its constituents. These attain their maximum visibility when the plane of analysation is at 45° to these planes, and vanish when the plane of polarization is parallel to either of the principal planes.

The phenomena of chromatic polarization afford a ready means of detecting doubly refracting structure in cases, such as that produced in isotropic bodies by strain, in which its effects are very minute. Thus a bar of glass of sufficient thickness, placed in the diagonal position between a crossed polarize and analyser and bent in a plane perpendicular to that of vision, exhibits two sets of coloured bands separated by a neutral line, the double refraction being positive on the dilated and negative on the compressed side. Again, a system of rings, similar to those of an uniaxal plate perpendicular to the axis, may be produced with a glass cylinder by transmitting heat from its surface to its axes by immersion in heated oil, and glass that has been raised to a red heat and then cooled rapidly at its edges gives in polarized light an interference pattern of a regular form dependent upon the shape of the contour.

Rotary Polarization.—In generala stream of plane-polarized light undergoes no change in traversing a plate of an uniaxal crystal in the direction of its axis, and when the emergent stream is analysed, the light, if originally white, is found to be colourless and to be extinguished when the polarize and analyser are crossed. When, however, a plate of quartz is used in this experiment, the light is coloured and is in no case cut off by the analyser, the tint, however, changing as the analyser is rotated. This phenomenon may be explained, as D. F. J. Arago pointed out, by supposing that in passing through the plate the plane of polarization of each monochromatic constituent is rotated by an amount dependent upon the frequency—an explanation that may be at once verified either by using monochromatic light or by analysing the light with a spectroscope, the spectrum in the latter case being traversed by one or more dark bands, according to the thickness of the plate, that pass along the spectrum from end to end as the analyser is rotated. J. B. Biot further ascertained that this rotation of the plane of polarization varies as the distance traversed in the plate and very nearly as the inverse square of the wave-length, and found that with certain specimens of quartz the rotation is in a clockwise or right-handed direction to an observer receiving the light, while in others it is in the opposite direction, and that equal plates of the right- and left-hand varieties neutralize one another’s effects.

A similar rotary property is possessed by other uniaxal crystals, such as cinnabar and the thiosulphates of potassium, lead and calcium, and as H. C. Pocklington (Phil. Mag., 1901 [6] ii. 361) and J. H. Dufet (Journ. de phys., 1904 [4], iii. 757) have shown by a few biaxal crystals, such as sugar and Rochelle salt, the rotation produced by a given thickness being in general different, and in some cases of opposite sign for the two optic axes. Further, certain cubic crystals, such as sodium chlorate and bromate, and also some liquids and even vapours, rotate the plane of polarization of the light that traverses them, whatever may be the direction of the stream.

In crystals the rotary property appears to be sometimes inherent