polarized light being transmitted from the collimator by the auxiliary polarizer. After obtaining this setting, the circle of the prism table is rotated through 180°, so that the auxiliary nicol is reversed end for end, and a new setting of the analyzer is made. It is clear that the difference between the settings taken before and after the reversal is double the small deviation of the principal plane of the polarizer from the vertical axis. Consequently, a setting midway between these settings will yield a match on plane polarized light only when the principal plane of the polarizer contains the vertical axis. The auxiliary nicol is therefore removed and the collimator polarizer is replaced in its circle and rotated until the match for that setting of the analyzer is accomplished. This gives the polarizer setting on its circle for zero azimuth of its principal plane, and with this setting known, any other desired azimuth can be obtained to the degree of precision afforded by the sensitivity of the halfshade system and the analyzer and polarizer circles.

Polarization by reflection at the polarizing angle of incidence from specular surfaces on transparent materials has also been used [10, 11] to determine the position of the polarizer for zero azimuth. When the reflecting surface of the glass alining prism or its equivalent contains the vertical axis and its polarizing angle of incidence is known, the setting of the analyzer corresponding to zero azimuth of polarizer is easily obtained by an analyzer match on light reflected at that incidence. For this test the incident monochromatic light may be either polarized or natural; but if it is plane polarized, the azimuth of the principal plane of the collimator nicol with respect to the plane of incidence must not be too small, since the intensity of the reflected light will then be so low that a match setting of the analyzer is impossible.

When the polarizing angle of the reflector is unknown, it, and also the zero azimuth, may be determined with a precision approximating that of procuring an analyzer match if a series of observations is made for two or more azimuths of the polarizing nicol and at two or more angles of incidence for each azimuth. The azimuths (not exactly known) of the polarizer's oscillation plane with respect to the plane of incidence should be equally distributed above and below zero but not so near that observations are difficult. The angles of incidence should be chosen in about equal numbers on each side of the only approximately known polarizing angle and should not depart from it by more than a few degrees except in preliminary observations. When the observations (angles of incidence and analyzer settings for match) are plotted, the curves (almost straight lines for a narrow range near polarizing incidence) will intersect at the polarizing angle, and the analyzer reading corresponding to this incidence is the analyzer setting for matching on light polarized in the plane of incidence.

Although a graduated collimator circle for the polarizer is advantageous, it is obvious that none of these methods require this, since all necessary azimuth readings may be referred to the analyzer circle. Moreover, once the analyzer setting for light polarized in the plane of incidence is obtained, the polarizing spectrometer may be moved about and used to set the principal plane of polarizers in other instruments with respect to vertical, provided the spectrometer is supplied with adequate leveling devices, which assure a coincidence of its axis with that direction.

10. PRODUCTION OF ELLIPTICALLY POLARIZED LIGHT

(a) BY DOUBLY REFRACTING PLATES

Doubly refracting plates for the production and compensation of elliptically polarized light can be prepared from either uni- or bi-axial crystals. Strained plates of isotropic materials, since they show the so-called "accidental double refraction," are also often used in polarimetric instruments instead of crystalline plates.

Doubly refracting plates from uniaxial crystals are usually cut parallel to the optic axis. Consequently, a beam of plane polarized light normally incident on such a plate is, in general, resolved into two undiverging plane polarized components, the extraordinary with its oscillation plane parallel to the optic axis (X- or Z-axis, depending upon whether the crystal is optically prolate or oblate) and the ordinary with its oscillation plane parallel to the optic normal.

If the plate is from an optically oblate crystal, the extraordinary component traverses it with the greater velocity, Me<Mo. If by convention the direction of the faster oscillation is chosen as the reference direction in the plate, the optic axis (X-axis), parallel to that direction in this case, should for convenience be marked "fast." In the case of plates from optically prolate crystals, >, and the optic axis (Z-axis) is the direction of the slower oscillation and should be marked "slow."

If the amplitude of the incident rectilinear oscillation is "a", and its azimuth with respect to the fast axis of the plate is y, the amplitudes of the components in, and perpendicular to, that axis are a cosy, and a sin, at incidence. Neglecting loss by absorption and reflection, the amplitudes are unchanged on emergence from the plate, but a phase difference proportional to the plate thickness, D, will have been introduced between the oscillations, which were obviously in phase at incidence. According to eq 3, and since z=D, this phase difference, 8, 2πD(μo—μe)/λ=d, (in an optically oblate plate, for example), and in eq 5 to 9, it is the equivalent of 28. Moreover, according to identities of the preceding eq 8, the ratio of the amplitudes a sin la cos y=tan y=tan . When 8, and are known, the characteristics of the resultant elliptical oscillation may be determined from eq 9.

To determine 8, with the needed accuracy usually requires some precise method of calibration, but its approximate value can be computed from D and the refractive indices if the crystal and the manner in which it was sectioned to produce the plate are known. Moreover, since the difference between the indices increases in general with decreasing wave length, it is usually necessary to determine 6, at several points in that portion of the spectrum in which the plate is to be used.

Such doubly refracting plates are used chiefly as accessories for polarizing microscopes and for such polarimeters as are used in measurements on elliptically polarized light. These accessory plates are usually rated in terms of the phase difference (or relative retardation) which they introduce between components having some designated wave length.

Although the relative retardation, especially when small, is more commonly expressed in circular degrees (or possible radians), the equivalent number (N) of wave lengths may also be used to designate

the "power" of a plate. For example, a plate having a thickness such that it causes a relative retardation of 360° (2 radians) between components having a wave length X' may be termed "a wave plate for '," or if the relative retardation is only 90°, the term "quarterwave plate" is generally employed. When the relative retardation is even smaller, the terms "10° elliptic compensator" or "4° elliptic halfshade" (as examples) are often used to designate not only the power of the plate but also its purpose.

Serviceable doubly refracting plates may be made from almost any sufficiently large, transparent, anisotropic crystal, provided the difference in the refractive indices for the plane polarized component beams is not so great that a plate producing the required relative retardation is too thin and fragile. For example, a wave plate cut from calcite parallel to the optic axis would be quite thin, and since N=8/2=1 for a wave plate, the thickness can be computed from the relation D=X/o-He. For a calcite plate cut in such a manner, the difference in the indices is about 0.172 for sodium light, and consequently D is roughly 0.0034 mm. In the case of quartz (optically prolate) - 0.0091 approximately for sodium light, and for a corresponding wave plate cut parallel to the optic axis, D is roughly 0.065 mm.

Mica, a monoclinic (pseudohexagonal) crystal, is much used in making doubly refracting plates (especially those having very low relative retardations). This follows because its perfect basal cleavage, giving thin elastic sheets, makes it relatively easy to prepare fairly uniform plates of considerable area and of almost any needed thickness. The acute bisectrix (X-axis) of this mineral makes an angle with the normal to the plate (basal cleavage) surface that ranges between 0° and 2°, depending on the specimen, while the angle between the optic axes is about 70°. The Z-axis is consequently the oscillation direction of the "slow" component and the Y-axis that of the "fast", and the corresponding refractive indices may be used for computing the approximate thickness of mica wave plates. While these indices vary considerably with the specimen, their difference for sodium light is of the order 0.004, and D for a wave plate at that wave length is consequently about 0.14 mm. Accordingly, the thickness of a mica quarter-wave plate for sodium light is about 0.03 mm [5, p. 352], and of a 2° halfshade, about 0.0008 mm. Some of the thinnest mica halfshades used show the brilliant first-order interference colors by reflection in white light.

When designed for use as accessories, thin doubly refracting plates, or even thick ones not made of comparatively hard crystals, should be mounted in Canada balsam between glass cover plates, unless the nature of the measurements requires other mounting materials. The cover plates should obviously be free from strain, since otherwise the "accidental" double refraction modifies the power of the enclosed plates. This modification can be particularly disturbing, since it is seldom uniform over the plate aperture. Multipel reflection between the surfaces of the plate or of its covers is another factor which sometimes causes certain annoying modifications in the performance of the plate. In many cases the first of the multiple images may become brighter than the primary image as the "matching" of a halfshade field is approached, and a setting for "complete" extinction is impossible if the images coincide.

Tilting a doubly refracting plate with respect to the light beam changes the effective order of the plate. Consequently, when the measurement undertaken depends on previous or independent calibrations of such a plate, it is necessary to maintain the light beam normal to the plate unless the tilting is controlled and made a part of the measuring procedure. Finally, in very precise measurements by certain methods, it may be necessary to consider other factors (temperature effects, for example) which might affect the order of a plate to a degree not compatible with the desired precision.

(b) BY ELECTRIC AND MAGNETIC FIELDS

Besides doubly refracting crystals and strained media, metallic reflection has also been mentioned as producing marked elliptical polarization. Because of their theoretical significance rather than their practical importance in polarimetric measurements, mention should also be made of the elliptical polarization effects observed when media in magnetic and in electric fields are traversed perpendicular to the lines of force by polarized light. The Zeeman effect is another condition which should not be passed without mention, because in this particular case polarized light is apparently emitted by a source. The effect is observed when incandescent vapors emitting line spectra are placed in strong magnetic fields. Lines unpolarized under normal conditions of emission are so affected under the influence of the fields that they are resolved into polarized component lines by a spectroscope. In the simplest of many more or less complicated effects, there are two oppositely rotating circularly polarized component lines when the source is viewed along the lines of magnetic force, and three plane polarized component spectral lines when the viewing is at right angles.

As compared to the normal unpolarized spectral line, the component circularly polarized in the direction of the amperian current producing the magnetic field is decreased in frequency, the other is increased. On the same basis of comparison, the central plane polarized component (oscillation plane parallel to the magnetic force) is unchanged in frequency, while the other two (oscillation planes perpendicular to the force) suffer the same changes in frequency as the circularly polarized components.

11. MEASUREMENTS ON ELLIPTICALLY POLARIZED LIGHT

From eq 5 to 10 it is obvious that in order to determine the oscillation characteristics of a given beam of elliptically polarized light, it is necessary to measure two out of the four following elements of the representative ellipse. These determinative elements are the ratio (tan ) of the minor to the major axis; the azimuth (7) of the major axis with respect to some chosen reference plane; the ratio (tan) of the amplitudes of the rectangular rectilinear oscillations which are obtained when the elliptical oscillation is resolved into plane polarized components in and parallel to the same reference plane; and the phase difference (28) between such components. If the elliptical polarization is produced from a plane or elliptically polarized light beam by some agency such as a doubly refracting plate, it is obviously necessary to determine not only the characteristics of the emergent light but also those of the beam incident on the plate. Moreover, the azi

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muth (7) of the principal plane of that plate with respect to the reference plane must also be known. In general, therefore, there are five measurements which must be made. In addition, the location of the chosen reference plane may require several measurements, as already shown.

The apparatus for the determination of these various elements consists of a polarizer for producing the oscillation form of the incident beam and an elliptic analyzer capable of yielding such measurements as may be required to analyze the oscillation characteristics of both the incident and emergent beams. When combined, the resulting instrument is adapted for practically any required polarimetric measurements on a polarized beam with any definite oscillation form from plane to circular. Such an instrument is, therefore, essentially a universal polarimeter.

(a) ELLIPTIC POLARIZERS AND HALFSHADES

Very often the polarizer is simply a nicol prism. In such a case tan y=0 if y=0. However, the polarizer usually is set in such a position with respect to the principal plane of the doubly refracting plate under investigation that 1/4 or 3/4. In that case, this principal plane is the reference plane and y=0. Moreover, tan ¥1 = ±1· and 8,=0.

In some cases the polarizer consists of a nicol followed by a quarterwave plate, which is so set that it produces circularly polarized light, which may have either a right or left vector rotation. With this device, y, at incidence on a second plate, is obviously undeterminate, while Y1 = ;=π/4 and 28;=π/2.

Seldom is there any advantage in using incident beams that are other than plane or circularly polarized, unless an elliptic halfshade following the nicol is introduced. In such cases the incident beam is divided into two parts, with different ellipses representing their polarization. The axis ratios of these ellipses may be practically equal or different. When equal, the vector rotations in the two parts of the beam are opposite. When the ratios are different, that for one part of the beam is generally zero and that for the other is small. Usually, with such halfshades, the azimuths, y, for both parts of the beam are practically that of the light emerging from the polarizing nicol. Balanced halfshades produce the condition in which the axis ratios for the two parts of the beam are equal, and the Bravais [1, p. 348] biplate is an example. The Brace elliptic halfshade [12] is an example of the unbalanced type.

(b) ELLIPTIC ANALYZERS

The elliptic analyzer usually consists of an elliptic compensator and a following nicol prism, while in some cases an elliptic halfshade is introduced between these parts rather than in the polarizer. Moreover, the simple nicol is sometimes replaced by a split (halfshade) nicol. The use of both halfshades divides the field (or beam) into four parts which must be matched in intensity for an analyzer setting. Except during initial adjustments in the assembly of the analyzer, the elliptic halfshade is bound to the nicol and does not rotate independently. However, both the compensator and the nicol must be capable of independent rotation. A universal analyzer containing the three parts has been designed and described by Skinner [13].