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6. ROTATION OF THE PLANE OF POLARIZATION-PHOTOGYRIC EFFECTS

Most isotropic substances are "optically inactive," or agyric; that is, they do not normally rotate the plane of polarization when transmitting polarized light [4, p. 272]. However, a large number of substances are photogyric. Possibly a preponderance of the crystalline materials might also be included in the class of optically inactive substances, but such a statement cannot be made definitely because too often rotatory power of a crystallogyric material is masked by ordinary double-refraction effects. In the many substances which change the azimuth of the plane of polarization, the rotation is proportional to the length of the light path within them. Under certain influences, all substances may possess a rotatory power.

Substances naturally photogyric are said to be "optically active." Sugar solutions and essential oils are examples of isotropic substances which are photogyric. Some of these substances are known to retain the rotatory power in the amorphous solid state (sugar, tartaric acid, etc.) and also in the vapor state (turpentine oil, etc.). Quartz and cinnabar are well-known examples of crystallogyric substances, and the first of these is exceptionally important because of its extended use in polarimetric and other optical instruments. Even in the isometric, or cubic, system many crystals possess a weak rotatory power. Crystalline sodium chlorate is an example of these, and its rotatory power is the same without regard to the direction of the light path through it. Moreover, in spite of the masking effects of ordinary double refraction, it has been possible of late to show that some of the biaxial crystals also possess a rotatory power. Finally, all substances are magnetogyric, since they rotate the plane of polarization when they are placed in a magnetic field, and when the light path through them has a component parallel to the lines of magnetic force, the effect is that of ordinary double refraction. In this respect an ordinary medium in a magnetic field is somewhat similar to a uniaxial crystallogyric mineral.

Photogyric isotropic substances are isogyric, since they have the same rotatory power, independent of the direction of the light path through them, while many minerals, such as quartz, are anisogyric, since they appear to be "optically inactive" if the path direction is normal to the optic axis, although they rotate the plane of polarization if the path parallels that axis. These minerals are generally known to exist in two crystallographic forms, one of which is dextrogyrate and the other laevogyrate. These forms are also often referred to by other terms, such as "dextrorotatory," "right-handed," and "positive' crystals, in the first case, and by corresponding terms in the second. The two forms often can be distinguished by casual inspection, and in quartz, for instance, the form may be indicated by the oblique striations on certain surfaces if the natural faces are reasonably intact [5, p. 571]. These striations show that the crystals are plagihedral, and if the striations lean to the right when they face the observer and if the apex of the adjacent pyramid is upward, the crystal is dextrogyrate [6, p. 31].

A dextrogyrate substance rotates the plane of polarization in a clockwise direction to an observer looking in the apparent direction of the source and at the clock face. That is, according to the previ

ously chosen convention, it produces a negative rotation, although in connection with these photogyric effects it is usually called a positive one. Obviously, the direction of the rotation produced by a laevogyrate material is the reverse. Reversing the direction in which light traverses these media does not alter the direction of the rotation. For this reason the rotatory effect is nullified if the light is reflected backward over its path through a naturally rotatory material. In this respect, magnetogyratory effects are different, since these rotatory effects are doubled when the light is returned by reflection over its path through a material situated in a magnetic field of unchanged direction. In other words, if the direction of the light is unchanged and the magnetic field is reversed, the direction of the rotation is also reversed. Consequently, the direction of the magnetorotation in a material must be defined with respect to the direction of the magnetic field and not with respect to that in which the light travels. If the light, in traversing a material in a magnetic field, passes from the south-seeking to the north-seeking pole of the magnet producing the field, the magnetogyric effects, especially in most diamagnetic materials, are usually dextrogyrate according to the photogyric convention. In other words, the effect is apparently more often than not dextrogyrate whenever the light travels opposite to the direction of the magnetic force. From this it appears that the direction of the magnetorotatory effect in diamagnetic substances (for example, glass) is usually that of the amperian current, which would produce the causative magnetic field if a solenoid about the light beam were used instead of a magnet. Dextrogyric and laevogyric effects (defined with respect to the magnetic field) are found, however, in both diamagnetic and paramagnetic media.

In all of these photogyric effects the rotatory power of the medium depends on the wave-length of the light. Usually the rotation in the visible spectrum increases with decreasing wave length, but there are many exceptions, and some media are laevogyric in one part of the spectrum and dextrogyric in another. This change may occur suddenly in the range of an absorption band or gradually in the region. between two such bands.

7. CIRCULARLY POLARIZED COMPONENT BEAMS IN PHOTOGYRIC EFFECTS

It has been shown experimentally that plane polarized light, normally incident on a plate of quartz cut perpendicular to the optic axis, is resolved into two equally intense circularly polarized component beams which are transmitted at slightly different velocities and have opposite vector rotations. For paths in directions between the optic axis and its normal, it has also been found that a plane polarized beam is resolved into two elliptically polarized components having different velocities. However, the axis ratios of the elliptical oscillations decrease very rapidly as the direction of the path departs from the optic axis.

Whenever plane polarized light, on entering a naturally or artificially photogyric material, is resolved into two equally intense circularly polarized beams having opposite vector rotations and different velocities in the material, it is obvious that at that instant the amplitudes of the oscillations are equal, (r1=r2), and the difference

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

απ α1
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,#2). 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, 0, is such that tan 0 μ (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 y 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,

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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 7/2-0', and from this incidence it also continually increases and becomes unity when 0=π/2. Since a'=0 when 0+0=π/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 After reflection, the azimuth (y") of the practically rectilinear reflected oscillation is obtained from the relation

=tan y.

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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 O to the polarizing angle, the rotation, ('-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 = 2.

π

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 , and after 0' (>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.

π

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 6, 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 π/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

1+2=sin @ tan 7.

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

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