in phase must be zero. As a consequence, the azimuths of the radius vectors with respect to the original plane of polarization are equal and opposite. When the light emerges from an "optically active" transparent plate, the amplitudes, only slightly changed, are again at least practically equal, but the difference in the velocities of the two component beams while in the plate has now introduced a relative difference in phase (dp) which, since there is no appreciable divergence effect, is proportional to the plate thickness, D. Consequently, the light, on the recombination of the circularly polarized components, is plane polarized at an azimuth

α2 απ
2

with respect to the plane of polarization of the incident beams; or y is the change in azimuth of the emergent oscillation plane with respect to incident oscillation plane.

If the subscripts 1 and 2 represent, respectively, counterclockwise and clockwise oscillations, it is apparent (since a, and a2 by convention represent positive retardations) that y represents a clockwise rotation of the oscillation plane (and also of the plane of polarization) whenever the component, circularly polarized in the clockwise sense, has the greater velocity while passing through the plate; that is, y is negative, since a1>a. This is in accord with the experimental results, which show that the magnetorotation in glass is right-handed (direction of amperian current) and that the clockwise oscillation is actually as well as relatively accelerated by the magnetic field.

Some photogyric absorbing media absorb the circularly polarized component beams differentially, and as a result the amplitudes on emergence are unequal (r,#r2). In such cases the recombined components produce an elliptically polarized resultant, as previously shown. This effect is usually designated as "circular dichroism."

8. POLARIZATION OF LIGHT BY REFLECTION

A specular surface on any transparent material (for example, glass and water) reflects light that at one angle of incidence is almost perfectly plane polarized in the plane of incidence. As discovered by Brewster, this occurs whenever the angle of incidence, e, is such that tan 0 (the refractive index of the material). Since μ varies with the light frequency, the polarizing angle of incidence also varies accordingly. For the best results in obtaining plane polarized light by this method, a monochromatic beam of parallel light rays and a good plane reflector are required. Even then surface films and other imperfections in the reflecting surface (which may be wholly invisible) often cause some incompleteness in the polarization. If the incident light is plane polarized and its plane of polarization and that of incidence are oblique, surface films cause the reflected light to be elliptically polarized to a very slight degree (that is, tan v is small).

This method of obtaining plane polarized light is very inefficient because only a small percentage of the incident light is reflected. To increase the efficiency, a pile of plates (several transparent plate reflectors in series) is used, but the results are still far inferior to those obtained with other devices. Moreover, the deflection of the re

flected beam causes so much inconvenience that this method is usually employed only in demonstrational and a few other nonprecision instruments.

Regardless of the angle of incidence or number of plates employed, the transmitted beam is never more than partially polarized if the incident beam is unpolarized. Although undeflected, the transmitted beam is, therefore, even less desirable than the reflected one as a source of polarized light.

At all oblique angles of incidence other than the polarizing angle, the reflected, as well as the transmitted, beam is only partially polarized, while at perpendicular and parallel incidence there are no polarizing effects. This naturally suggests that the components of the incident oscillations taken parallel to and perpendicular to the plane of incidence are reflected according to different laws. Such a difference is provided for by the Fresnel equations [3, p. 351], which express the ratios of the reflected amplitudes, a'' and b'', to the incident, a and b, (X- and Y-direction respectively parallel to and perpendicular to the plane of incidence) in terms of functions of the angles of incidence, e, and refraction, e'. That is,

Obviously, the square of the second ratio increases continually from an indeterminate value to unity as increases from 0 to π/2, while the square of the first ratio decreases from the same indeterminate value to 0 as increases from 0 to π/2-0', and from this incidence. it also continually increases and becomes unity when 0=π/2. Since a" 0 when 0+0=/2, it follows that the reflected light is then plane polarized with its oscillation direction parallel to the plane of the reflecting surface. Moreover, by making use of the law of refraction, sin '/sin =μ, Brewster's law for the relation between the polarizing angle and the refractive index may be derived.

When approaches /2 (grazing incidence), the amplitude ratios approach unity, the limiting value for no reflecting surface. At normal incidence, 0=0, the ratios are equal (disregarding sign) because the oscillation directions of both components are parallel to the mirror surface. If the incident light is plane polarized and the azimuth of its oscillation direction (X-axis in plane of incidence) is y, then b/a=tan =tan y. After reflection, the azimuth (7") of the practically rectilinear reflected oscillation is obtained from the relation

Thus at normal incidence and to an observer who always looks in the direction (-Z) of the source both before and after incidence (the source being, respectively, real and apparent), reflection rotates the plane of oscillation in a manner remindful of the "from right to left" perversion of a reflected image and y+y". As @ increases from 0 to the polarizing angle, the rotation, (y''—y), decreases to onehalf the value it had at normal incidence, because a" is then practically negligible. Moreover, since cos (0+0') changes sign at the polarizing

angle, the rotation continues to decrease and becomes zero when 0=T/2.

The effect of the change in sign of cos (0+0') is the same as that which would be caused by a sudden change from to 0 in a phase difference between the reflected X- and Y-components (that is, between the oscillation corresponding to a" and b') as e increases through the polarizing angle. Moreover, if the mirror surface is of such a nature that the light reflected near the polarizing angle is elliptically polarized, intermediate phase differences are actually observable, and any change in phase is not abrupt, since it develops more or less gradually as incidence increases and reaches the value /2 at or near the incidence 0=arctangent u. (The sign of the phase difference is not taken into consideration.) Normally, the angle of incidence corresponding to a phase difference of π/2 would be considered as the polarizing angle, since it is then that the major axis of the oscillation path is parallel to the mirror surface. Obviously, the Fresnel equations in the simple form presented above must be modified for those cases in which the reflected (and transmitted) light is elliptically polarized. The similar expressions developed for the relation between the components in the case of total reflection (a phenomenon which occurs only when M>, and after '(0) reaches /2) are examples of a relatively simple modification [3, p. 358]. Similar expressions are also developed for reflection from metal surfaces. In the development of the equations for such cases, the angles of refraction are treated as imaginary or complex qualities.

In metallic reflection, the incident plane polarized light not absorbed (some light may be transmitted if the mirror is very thin) is reflected as elliptically polarized light at all angles of incidence except normal and grazing. The particular angle of reflection (and incidence), termed the "principal incidence," , corresponds to the polarizing angle of transparent reflectors in that it is defined as the angle of incidence for which the phase difference between the components in and normal to the plane of incidence is π/2. Thus at this incidence the major axis is parallel to the reflecting surface. The angle having a tangent equal to the ratio of the axes (B/A) of the reflected elliptical oscillation which corresponds to this incidence is named the "principal azimuth,"V, provided the azimuth, y, of the incident plane polarized monochromatic light is /4 (or 3/4). When these angles are determined for a metallic mirror, the refractive index, μ, and the extinction coefficient, x, may be closely approximated in many cases by computing them from the simplified equations [1, p. 363]

K=tan 2

uv1+r=sin 8 tan .

Ordinarily the characteristics of the reflected elliptically polarized light are determined at other angles of incidence than . The equations [4, p. 261] for determining μ and ke in these cases are reduced to the forms

and require the measurement of (the arctangent of the ratio (b"/a") of the reflected components normal to and in the plane of incidence) and 28 (the phase difference introduced between the components on reflection). Moreover, it is assumed, as above, that the incident light is monochromatic and polarized in a plane having the azimuth Y1=π/4=1. In consideration of relations (a), (b), and (e) of eq 9, these equations may also be written in terms of functions of the ratio of the axes of the reflected ellipse and the azimuth of its major axis.

The coefficients, μ and Ke, determined experimentally at different angles of incidence, will vary with that angle. The relation between coefficients, and Ko, for normal incidence to those for any other incidence may be simplified greatly by neglecting all squares of higher

ко

are adequate in computing the coefficients at normal incidence. As in the reflection from transparent media, surface films are disturbing factors, and care must be taken to eliminate them as far as possible if the optical coefficients of metals are determined by this catoptric method. Under the best conditions, the method does not yield results with an accuracy comparing to that of results obtained by dioptric methods, but it can be used in cases where they are practically inapplicable.

9. SEGREGATION OF PLANE POLARIZED LIGHT BEAMS

(a) POLARIZERS FROM LARGE CRYSTALS

While a practically plane polarized beam of light is easily obtained by reflection from mirror surfaces on transparent media, the method is, as already stated, inefficient and unsuited for most polarimetric measurements. The resolution of natural light into two equally intense plane polarized beams by double refraction serves, on the contrary, as an ideal method of producing polarized light whenever it is possible to segregate one of the component beams without undue loss of intensity. In some cases this segregation is accomplished to a degree by the crystal itself. Tourmaline, for example, not only resolves natural light into plane polarized beams but also absorbs them

differentially. As a result, a plate of no great thickness (possibly 1 or 2 mm, depending on the crystal) transmits very little more than the extraordinary beam. The oscillation direction in the extraordinary beam lies in the plane containing both the normal to the plate surface and the optic axis. This plane, since it is normal to the face of the plate, is termed the principal section of the refracting surface. Obviously two such plates in series with their principal planes at right angles will almost completely absorb both components, since the extraordinary of the first becomes the ordinary beam of the second.

Polarizers of this sort are not efficient because the transmitted component is also reduced materially in its intensity. Moreover, the differential absorption varies with wave length, and when complete polarization is almost attained with a minimum loss of intensity for one color, the intensities of other colors may in comparison be greatly reduced or the polarization for other parts of the spectrum may be far from sufficient. Similar disadvantages are found in certain artificial polarizers.

Crystals such as quartz and calcite are highly transparent for both beams over a very great range of wave lengths. Consequently, the production of an efficient polarizer requires only the segregation of one of the polarized components by some artificial means which does not materially reduce the intensity of the component. The comparative frequency with which sufficiently large crystals of quartz and calcite are found has caused these materials to become very important adjuncts in all polarimetric work. The smaller divergence between the refractive indices of the component beams in quartz has limited the use of this material in the production of simple polarizers for visible light. However, the difference between the calcite indices is sufficient in some directions through the crystal to cause a comparatively large deviation of the ordinary and extraordinary beams. This large divergence makes it easy to segregate either of the plane polarized beams. In fact, such a beam of plane parallel light may be obtained with a cross-sectional diameter equal to one-tenth the crystal thickness simply by the proper use of diaphragms. Obviously, the procurement of beams having the cross section (or aperture) often required would necessitate the use of unduly large crystals. Consequently, Nicol took advantage of the differences between the refractive indices for the two beams to effect the total reflection of the ordinary beam at an interface formed by a cementing material with an intermediate index, and he thus devised a type of prism which greatly increases the aperture obtainable at the expense of a comparatively slight loss of intensity. Of the various modifications of this prism, some have so nearly the form of the original cleavage crystal that the direction of the optic axis, the principal plane (section) of the faces, and consequently, the direction of the oscillation may be determined approximately by simple inspection.

In a calcite cleavage rhombohedron, the X-direction is the optic axis and makes equal angles with the three edges at either of the two fully obtuse corners formed by the cleavage surfaces. These edges include three facial angles of about 101°55' at these corners, and the optic axis makes an angle of about 63°44′ and 45°23' with each edge and face, respectively. When light is incident normally on a face, the ordinary ray (direction of the ordinary beam) passes directly through