From this it is clear that the amplitudes, A, and A2, are periodic functions of the time, and that consequently the amplitude a is also. The frequency of this variation is, however, comparatively small, since is much greater than w'. Although w is constant, the variability of a is really the equivalent of a variable frequency in the resultant. This simple example indicates, therefore, that both amplitude and frequency of an oscillation, which is the resultant of an infinitude of homogeneous elementary wave trains with slightly different frequencies, must be variable. At least this must be the conclusion if it is permissible to assume that any such large number of elementary trains as is required to produce a spectral line can be divided into two parts which yield unrelated resultants with slightly different frequencies that approximate those of the components in the above example. Thus if by any chance or because of any reason these partial resultants possessed amplitudes and frequencies that varied little during intervals equivalent to many thousand periods of the elementary trains, their resultant would nevertheless possess all of the frequency and amplitude variations shown during a similar interval by the above simple example of a resultant. Moreover, this conclusion applies to the two partial resultants themselves after due consideration is given to the probable condition that the differences between the frequencies of their components (partial resultants restricted to still smaller spectral ranges) would presumably be smaller. If, in the case of rectangular components, it is further assumed that the number, amplitudes, and phases of the elementary trains contributing to the y- and to the x-resultant oscillations are different, it is obvious that there will be no relationship between the variabilities appearing in either the amplitudes or the frequencies of the two resultants. Hence, rectangular component beams obtained from an unpolarized beam will not be coherent, if they are resultants of that

nature.

If the oscillations of the elementary trains contributing to the resultants were to have constant and equal amplitudes and frequencies, so that the resultant amplitudes and frequencies would be constant, as earlier assumed, it would follow that such trains must remain unbroken and unchanged for time intervals at least as long as the intensity of the resultant light beam appears to be unvarying. Such a condition is undoubtedly contrary to that which actually exists, since the innumerable oscillators that originally contribute the elementary trains from a radiant and so-called monochromatic source are, compared to the time required to obtain a simple polarimetric observation, extremely short-lived, even though each may oscillate many thousand times. Consequently, during each oscillation period of the resultant, the contributions from a small portion of the oscillators cease and are

replaced by others from freshly excited oscillators. In general, the number, oscillation forms, phases, and to some extent the frequencies of the replacements during a period, will differ from those of the replaced contributions, although between averages taken over long time-intervals these differences are not apparent. Moreover, the contribution from each oscillator may change its oscillation form, etc., more or less gradually throughout the time interval in which it emits. In response to these variations in the number of contributing oscillators and in the oscillation form, etc., of the contributions, the resultant at any point in a light beam continually and compatibly alters its oscillation characteristics, regardless of the manner in which the contributions arrive at that point from the source. That is, even in light from the nearest possible approach to a monochromatic source, the resultant oscillation will vary in a random manner through all elliptical forms from rectilinear to circular without favor, and the direction of the light vector at its maximum during each oscillation will vary from period to period without favoring any azimuth. Moreover, the maximum magnitude of the vector during a period may on occasion approach zero if the amplitudes and phases of the elementary trains are such that almost complete interference occurs in some segments of the resultant train. Consequently, both the amplitude and frequency of any rectilinear component obtained from the resultant will also vary continually in a random manner approaching that previously suggested, and these variations in rectilinear components (taken at right angles, for instance) will be unlike. That is, such components of natural light are ordinarily incoherent.

That this incoherence must exist is perhaps clearer when it is considered that in general an oscillator contributes unequally to the components and that this inequality varies with the oscillator. Only when a rectilinear component of unpolarized light is further resolved into two or more rectilinear (or elliptical) components does each oscillator distribute its contribution coherently between the components and also in the same relative degree (as the other oscillators) to any one component. Therefore, such secondary components are coherent as a result of artificial manipulation.

The oscillation changes, contributed to a resultant by such an infinitude of oscillators with almost identical frequencies, develop quite slowly in comparison with the rapidity of primary oscillation. This comparative slowness results because the changes in the characteristics of the emission of an individual oscillator develop gradually and also because the number of replacements per period of the primary oscillation is small when compared to the very great number of simultaneously contributing oscillators. It is obvious, however, that on the average every one of this great number will be replaced within the lifetime of the youngest. Consequently, if one of a pair of secondary, coherent components that are plane polarized in different azimuths suffers a relative retardation corresponding to this time interval by being made to pass over a path sufficiently longer than that traversed by the other, the coherency of the two components will then no longer be evident, and the resultant light on their recombination will be unpolarized.

The effect of this incoherency appears not only in polarized light but also in the interference phenomenon of common light. That is, two incoherent beams (for example, beams from different parts of

some source) do not produce observable (so-called destructive) interference. Likewise, observable interference is never obtained when two primary plane polarized components of an unpolarized beam are brought into the same azimuth and recombined. This is the case even without introducing any path difference which would cause incoherence. Interference effects between unpolarized components, such as are produced from an unpolarized beam by an interferometer, occur because the component oscillations in any azimuth and in either component beam has its counterpart in the same azimuth in the other. Presumably, it will be only when the relative retardation between such beams becomes approximately equivalent to the lifetime of an oscillator that all observable interference vanishes. By using light from a suitable spectral-line source, interference effects have been observed until the difference in path exceeded a million wave lengths. This difference is of the order of 50 cm in air and corresponds roughly to a time interval of one 600 millionth of a second. Polarimetric measurements, however, are seldom concerned with retardations which exceed a few wave lengths.

3. COMPOSITION OF RECTILINEAR OSCILLATIONS HAVING

DIFFERENT AZIMUTHS

The superposition of two coherent plane polarized beams not having the same azimuth yields, in general, elliptically polarized light; or if they are not coherent, unpolarized light. Ordinarily it is necessary to consider only such beams as are polarized at right angles. Equations 1a and 1b are representative of the component oscillations occurring at any common point, z, in these beams. That is,

x=a cos (w1t-a) and y=b cos (w1t--ß).

If the conditions that b/a, wr, and w, are constant and that w2-w1=0 are approximated to a sufficient degree, the oscillations are coherent; otherwise, they are not. For the moment it will be considered that the coherency of the beams is a matter of doubt.

Assuming that resultant vector, r, of these oscillations is obtained in the same manner as the resultant in the composition of forces at right angles, it follows that

r2 = x2+ y2=a2 cos2 (w2t—a)+b2 cos2 (w1t—ß).

By writing the identities (for comparison, see equation 4).

and then by performing the necessary simple trigonometric transformations, it develops that

r2=a2 cos2 (wt - A'+8)+b2 cos2 (wt-A'—8)

(6)

If the conditions for coherence are strictly met, A'-' is constant and eq 6 is the equation of an ellipse, of which A and B are the semiaxes. Furthermore, the magnitudes of the amplitudes, a and b, lie between those of the semiaxes. Consequently, the identity

may be written, and it develops later that y and y±7/2 are the azimuths of the semiaxes.

If the conditions for coherence are almost met but not to the degree essential in elliptically polarized light, A, B, 8, w, and y are then all functions of t, and B/A and y vary between 0 and ∞ and between 0 and 2, respectively. The resultant light is unpolarized in this case, and an indicator point tracing out the variations in the magnitude and direction of the vector, r, would describe a long succession of unrepeated patterns akin to a complicated series of almost elliptical Lissajous figures, in which no ratio of axes or azimuth of major dimension is favored. Moreover, the possibility that both the major and minor dimensions occasionally approach zero almost simultaneously cannot be excluded.

A similar pattern intended to represent the vector oscillations in white light is too complicated and irregular for the imagination. However, even in the case of a component, as obtained from a beam of white light by an interferometer, every infinitesimal portion of its spectrum is practically coherent with the same elementary spectral portion of the other component. (This is true provided the previously mentioned limitation as to the relative phase (path) difference is imposed.) Consequently, the so-called channeled spectra appear whenever white light is analyzed by a spectroscope after it has been passed through an interferometer. Analogously, two rectangular components of a plane polarized beam of white light, which are recombined after one has been subjected to a relative retardation of several wave lengths that increases with decreasing wave length, will yield a spectrum in which every spectral element is, in the general sense, elliptically polarized. That is, as the wave length decreases, the oscillation forms will pass again and again through a series ranging from rectilinear through all degrees of elliptic to circular polarization. Moreover, on passing through the circular form, the azimuth of the major axis will shift from the first to the second quadrant or in the reverse direction. Such a resultant on being passed through a properly oriented nicol will, when examined with a spectroscope, also yield a channeled spectrum.

Light from a so-called monochromatic source emitting a line of average breadth (although the narrowest possible line still encompasses an infinitude of infinitesimal spectral elements), presents no appreciable spread in oscillation form after being treated as suggested above in the discussion on white light. That is, the resultant vector oscillation, even when representing all frequencies in the line, very closely approximates a simple elliptical form. As time proceeds, however, the ratio of the axes and their azimuths will fluctuate slightly about average values and the major axis will occasionally approach zero. After a sufficiently large number of periods, the representative pattern

of this oscillation is consequently an elliptical disk (if such a representation may be allowed) which through variations in the denisty of its shading could be made to suggest the dimensions of the approximate ellipses occurring most frequently. Regardless of these variations, the individual ellipses, representing the resultant oscillations of practically coherent perpendicular components, are all so nearly closed and have so nearly the same form and azimuth that the deviations from a perfectly stable oscillation form are negligible as far as all ordinary problems of elliptically polarized light are concerned.

For practical purposes, therefore, it may be considered that 2w'=w2-w1 =0, and consequently that 28-8-a-2A. It also may be assumed that y and the ratios b/a-tan and B/A=tan y, are constants, although a, b, B, and A vary. (As a matter of convenience,

it will be considered that the functions representing these ratios are always positive.)

From the identities (d), (e), and (f) in eq 5 it is evident that

(A2 — B2)2= a++b+2a2b2 cos 48
(a2+b2) tan 28' (a2-b2) tan 28

a2b2 sin2 28-A2B2 or sin 28=AB/ab

(8)

The double sign in the last relation will ordinarily be neglected, since it merely shows that sin 28 may be either positive or negative. It can be shown by graphic methods, however, that the vector rotation is positive (counter-clockwise to one looking toward the source, or, that is, in the negative direction) or negative according as this sine is positive or negative. Moreover, it can be shown that the major axis lies in the first and third quadrants or in the second and fourth, according as the cos 28 is positive or negative. With respect to the position of the major axis it also appears from identity 7 that y lies between /4 and 3/4 or between 7/4 and +/4, according as ab. On proceeding further with the development of the relationships existing between the identities used in the previous pages, it may be shown that

Asterisks indicate those equations found in the Theory of Optics [4, p. 15]. However, it will be noted that Schuster used the identity 8-B-a, as in the discussion of eq 1a and 1b.

Certain of these relationsips are required in all measurements on elliptically polarized light, and it will be noted that after any two of