the angles have been determined experimentally, the remaining four may, in general, be computed.

T2=A2

In order to appreciate that y is the azimuth of the major axis, it will be noted that wt-A'+d'=0 when r2-A2, and that (y/x) r2-A, is the tangent of the azimuth of the axis A with respect to axis X. It then can be shown by the use of the previously developed relations that

YB

where is the azimuth of B, also measured from the X-axis. The direction of the rotation of the vector, r, may also be found from the change in its azimuth y, with time. Thus

or, as previously stated, the sign of the sine of the relative retardation determines the direction of the vector rotation when a and b are considered as always being positive.

=

From eq 10a, 10b, and (a), (b), and (f) of eq 9 it is evident that y=4, B=0, 8'-0, and A'-'-a-B, if d=0. That is, the light is plane polarized, and the direction of its oscillation lies in the first and third quadrants. Many other special cases, where 0<b/a < ∞ and has positive or negative values of various multiples of π/4, may be analyzed easily by means of eq 5 to 10b.

4. COMPOSITION OF CIRCULAR OSCILLATIONS

∞

If beams of really homogeneous light were obtainable, they would, in the general sense, always be elliptically polarized, and if their periods were identical, their resultant would also be elliptically polarized in the broader sense. Within limits, the same is true in the case of elliptically polarized and practically homogeneous real light as long as the beams are coherent. Any beam of plane polarized light may, as a matter of fact, be considered as the resultant of two or more coherent beams of elliptically polarized light. However, it ordinarily is necessary to consider only that case in which the components of an elliptically or plane polarized resultant are two in number, and it will be assumed now that the components are circularly polarized. [3, p. 443].

As shown by eq 6, representative oscillations of two such components may be defined by their vectors, r1 and r2, and the continually and uniformly changing vector azimuths (wt-a1) and 72= ±(wt-α2) where the double signs indicate that the rotations may be either positive or negative. The identities y='+y" and y= '-"are used to facilitate the following transformations. It will be noted that y' is the mean azimuth of the two vectors, while 2y" is

the angle between them. Assuming again that the resultant vector is found as in the composition of forces, it follows that

where is the azimuth of the resultant. By writing the additional identities

This angle, "", lies between the resultant and the mean azimuth of the component vectors and obviously becomes zero whenever r1= ̃1⁄2 or y′′=0.

If they

απαν 2

Since the component vectors may rotate in the same or opposite directions, it becomes necessary to consider the two cases. rotate in the same direction, y'=±(wt· Consequently, the resultant and its direction are r2=r+r2+2r1r2 cos (a2-a) and y=(wt

Under these conditions, the resultant light is always circularly polarized.

If the component vectors rotate in opposite directions, y'

=

απ α

2

and y" = ± (wt α) + α1). ). Moreover, by substituting A2 + B2

2

and A2-B2 for 2 (+) and 4r, r2 and using eq 11, the resultant and its azimuth are given by the expressions r2=B2+(A2— B2) cos2

(wt-12) (see eg 6) and y= y'", where y'" is the azimuth

απ παι
2

of the resultant with respect to the major (or minor) axis of an ellipse representing the variations in r. Under such conditions, the resultant light is always elliptically polarized if r1#r2. Moreover, the azimuth of the major axis is that of vector r when wt=(a+a2)/2 or, what is the same, when y" (and, consequently, " also) becomes zero. The azimuth, YA, of the major axis is therefore (a2-a)/2. If the component vectors not only rotate in opposite directions but also are equal (r1=r2), it is apparent that B=0 and y'"=0. 02-α That is, in the

αι

Therefore, r=A cos (wt a1+α) and 7=

500)

2

αντα
2

composition of circular components with different directions of rotation, the resultant light is plane polarized in an azimuth determined. alone by the phase difference between the components. The roles played by the differences in amplitude and phase are, therefore, the reverse of those played by the same differences in the composition of rectilinear components.

5. POLARIZATION BY DOUBLE REFRACTION

As already stated, natural light may be considered as being the resultant of two or more incoherent component beams of light that are plane polarized in different azimuths. Moreover, when the light is actually resolved into components they are also found to be incoherent, and, if their planes of polarization are not exactly brought into a common azimuth, their recombination again results in unpolarized or partially polarized light. However, polarized light can be obtained from natural light by resolving it into plane polarized components and then rotating the polarization planes to a common azimuth before the components are recombined, by diverting the components into individual paths which are so divergent that recombination does not occur, or by employing some other means, such as the nicol prism, which eliminates all but one plane polarized component. Plane polarized light is easily produced by any of these methods, but circularly and elliptically polarized light are not usually obtained so directly from natural (unpolarized) light.

In general, the resolution of unpolarized (also of polarized) light into two polarized component beams occurs naturally whenever a beam of light traverses any doubly refracting (anisotropic) crystal. That is, a beam of light on entering such a crystal is resolved into two differently refracted beams, which on emergence are slightly separated and, ordinarily, plane polarized at right angles, except when they parallel certain unique directions in the crystals. The difference in the refraction for the two polarized component beams is, in certain selected directions in a number of crystalline minerals, large enough to cause a divergence of the beams that is sufficient to allow the complete elimination of one beam to be accomplished in various ways.

If the incident light is circularly polarized or unpolarized, the differently refracted component beams are of equal intensity; but if it is plane or elliptically polarized their relative intensity varies as the crystal is rotated about the beam as an axis. If plane polarized light is normally incident on a crystal face, the component beams (transmitted by a calcite crystal, for example) are equal in intensity only when the plane containing both refracted beams has an azimuth Y==π/4 with respect to the oscillation direction of the incident light. The ratio of the intensities for any azimuth is that of the squared sine and cosine of that angle or, if the amplitude of the incident light is a, the corresponding amplitudes of the components are a siny and

a cos Y.

Well-known examples of doubly refracting crystals are tourmaline, calcite, and quartz. These examples belong to the hexagonal (or the trigonal according to some classifications) system and to the class of uniaxial minerals, a class which also includes crystals of the tetragonal system. In optical terminology, quartz crystals are "positive" (or optically prolate) [4, p. 168]. Tourmaline and calcite are both "negative" (optically oblate). These terms are concerned with the relative velocities with which the component beams traverse the crystals in various directions. The wave surface (any surface or surfaces of equal phase) about a point source of light in an isotropic medium is spherical, since there is but one velocity regardless of direction. In a doubly refracting crystal, however, there are two concentric wave

323414°-42- -3

surfaces and, in uniaxial crystals, one is always spherical while the other is spheroidal. The component beam corresponding to the spherical wave surface traverses the crystal with the same velocity regardless of the path direction, and on entering (or leaving) the crystal obeys the normal laws of refraction. The velocity of the other component beam, as indicated by the spheroid, varies with the direction of the path. In general, a refracted beam corresponding to the spheroid does not lie in the plane containing both the incident beam and the normal to the surface of incidence. The two differently refracted component beams are therefore named, respectively, the ordinary and extraordinary, and, in optically prolate (positive) crystals, the velocity of the former is greater (index of refraction is less) than that of the latter beam, while in optically oblate (negative) crystals the reverse is true. That is, in optically positive crystals, the prolate spheroid representing the wave surface for extraordinary beams lies within the sphere representing a wave surface with the same phase for ordinary beams. In the case of a positive uniaxial crystal, the spheroid and sphere may be conceived of as having been generated by rotating an ellipse and its circumscribing circle about their common diameter, the direction of the major axis of the former. In a negative uniaxial crystal, the oblate spheroid encloses the sphere, and the axis of rotation of the generating ellipse and circumscribed circle is again the common diameter to both, but the direction of the minor axis of the ellipse. In both cases the direction of this common diameter is that of the optic axis and of the vertical crystallographic axis (C-axis) of the crystals.

If the wave surfaces for a uniaxial crystal are actually tangent at the points of intersection with their axis of rotation, two component beams traversing the crystal parallel to this axis will possess identical velocities, obey the ordinary laws of refraction, and will not diverge. Consequently, an incident beam of parallel light (polarized or unpolarized) will, after traversing a plate of such a crystal parallel to its optic axis, be unchanged in its polarization characteristics. In all other directions through the crystal the velocities are different, and the component beams are plane polarized at right angles and diverge.

Certain uniaxial crystals (for example, quartz) rotate the plane of polarization (or major and minor axes) when they are traversed parallel to their optic axis by a beam which at incidence is plane (or elliptically) polarized. This has been interpreted as showing that light, traversing these crystals even in a direction parallel to their optic axes, is resolved into components which have slightly different velocities and that sphere and spheroid do not touch [5, p. 580] at their intersections with the mutual axis of generation. It has been shown, moreover, that these components are circularly polarized; or, if the waves travel obliquely to the optic axis, the component beams are elliptically polarized.

The wave surfaces of a uniaxial crystal are a special case of a fourthdegree surface, which is represented analytically by the expression (x2+ y2+z2) (a2x2+b2y2+c2z2)—a2 (b2+c2) x2-b2 (c2+a2) y2-c2 (a2+b2) z2+a2bc2=0, in which it may be assumed that a>b>c [3, p. 335]. Obviously, if these constants are equal, the expression reduces to that for the single spherical wave surface of an isotropic medium (noncrystalline materials and also many crystals of the cubic system). If b=c only, the surface becomes a sphere within an oblate ellipsoid; or,

if b= a only, it is a prolate ellipsoid within a sphere. These special cases apply to the so-called "negative" and "positive" uniaxial crystals, respectively. If there is no equality between the constants, neither surface is spherical, and they represent the wave surfaces of biaxial crystals. The three sections of these surfaces by the three principal oscillation planes (XY, YZ, and XZ), obtained by letting z, x, and y, respectively, equal zero, are correspondingly a circle lying within, without, and neither wholly within nor wholly without an ellipse. In the last case the circle obviously cuts the ellipse at four points. The diameters joining opposite intersections give the directions of the axes of single-ray velocity, which make only a small angle with the two respective optic axes of the crystal. The angles between these diameters and also those between the optic axes (axes of singlewave velocity) are bisected by the coordinate directions, X and Z. These bisectors are termed the "acute" and "obtuse bisectrices,” according as the bisected angles are acute or obtuse. The crystal is said to be optically negative when X, and positive when Z, is the acute bisectrix. The XZ-plane, since it contains the optic axes, is known as the optic plane, and the Y-direction as the optic normal, of biaxial crystals. In the case of uniaxial crystals it may be considered that the optic axes and acute bisectrix coincide and result in a single optic axis. Consequently, these crystals have no definite optic plane, and the positions of the perpendicular obtuse bisectrix and optic normal about the optic axis are indeterminate.

In uniaxial crystals the optic axis coincides with the crystallographic C-axis (the so-called vertical axis), but in biaxial crystals neither of the optic axes coincides with a crystallographic axis nor, in general, do the coordinate or oscillation axes, X, Y, and Z. That is, in the orthorhombic system, the three mutually rectangular crystallographic axes, A, B, and C, coincide with the oscillation axes, although the disposition of the coincidences varies with the crystal; in the monoclinic system one of the oscillation axes (either X, Y,or Z) coincides with the orthodiagonal axis, B, and the other two without a coincidence lie in the plane of the mutually oblique A- and C-axes; while in the triclinic system an oscillation axis seldom actually, or even approximately, coincides with one of the crystallographic axes, which are all mutually oblique.

As a generalization for both uniaxial and biaxial crystals, it may be stated that when a light beam traverses them along either the X-, Y-, or Z-axis (except the optic axis in uniaxial crystals) it is resolved into two undiverging unequally retarded plane or eliptically polarized rectangular component beams, and that ordinarily the beam, when traversing them in any other direction (except those of the optic and single-ray axes in biaxial crystals), is similarly resolved, except that the components diverge. In uniaxial crystals, however, one component always follows the ordinary laws of refraction, while in biaxial crystals neither of the component beams necessarily does so.

With regard to the above indicated exceptions, a beam paralleling the optic axis is in some uniaxial crystals resolved into two unequally retarded circularly polarized components, while the optic and singleray axes of biaxial crystals are directions associated, respectively, with internal and external conical refraction. Also, some uniaxial (and possibly some biaxial) crystals resolve nonaxial beams into eliptically rather than plane polarized components.