maximum and minimum polarization of the light. In the particular
case in which a=b and α−β = ± (2n+1) π/2, the
vibrations are circular and the light is said to be circularly
polarized.
These different types of polarization may be obtained from a plane polarized stream by passing it through a quarter-wave plate, i.e. a crystalline plate of such a thickness that it introduces a relative retardation of a quarter of a wave between the component streams within it. Such plates are generally made of mica or selenite, and the normal to the plane of polarization of the most retarded stream is called “the axis of the plate.” If this axis be parallel or perpendicular to the primitive plane of polarization, the emergent beam remains plane polarized; it is circularly polarized if the axis be at 45° to the plane of polarization, and in other cases it is elliptically polarized with the axes of the elliptic path parallel and perpendicular to the axis of the plate. Conversely a quarter-wave plate may be employed for reducing a circularly or elliptically polarized stream to a state of plane polarization.
Two streams are said to be oppositely polarized when the one is, so far as relates to its polarization, what the other becomes when it is turned through an azimuth of 90° and has its character reversed as regards right and left hand. An analytical investigation of the conditions of interference of polarized streams of the most general type leads to the result that there will be no interference only when the two streams are oppositely polarized, and that when the polarizations are identical the interference will be perfect, the fluctuations of intensity being the greatest that the difference of intensity of the streams admits (Sir G. G. Stokes, Math. and Phys. Papers, iii. 233).
It remains to consider the constitution of common unpolarized light. Since a beam of common light can be resolved into plane polarized streams and these on re composition give a stream with properties indistinguishable from those of common light, whatever their relative retardation may be, it is natural to assume that an analytical representation of common light can be obtained in which no longitudinal vector occurs. On the other hand a stream of strictly monochromatic light with a polarization-vector that is entirely transversal must be (in general elliptically) polarized. Consequently it follows that common light cannot be absolutely monochromatic. The conditions that are necessary in order that a stream of light may behave as natural light have been investigated by Sir G. Gabriel Stokes (loc. cit.) and by E. Verdet (Oeuvres, i. 281), and it may be shown that two polarized streams of a definite character are analytically equivalent to common light provided that they are of equal intensity and oppositely polarized and that there is no common phase relation between the corresponding monochromatic constituents. Further a stream of light of the most general character is equivalent to the admixture of common and polarized light, the polarization being elliptical, circular or plane.
We see then that there are seven possible types of light: common light, polarized light and partially polarized light; the polarization in the two latter cases being elliptical, circular or plane. Common light, circularly polarized and partially circularly polarized light all have the characteristic of giving two streams of equal intensity on passing through a rhomb of Iceland spar, however it may be turned. They may, however, be distinguished by the fact that on previous transmission through a quarter-wave plate this property is retained in the case of common light, while with the two other types the relative intensity of the streams depends upon the orientation of the rhomb, and with circularly polarized light one stream may be made to vanish. Plane polarized light gives in general two streams of unequal intensity when examined with a rhomb, and for certain Positions of the crystal there is only one emergent stream. Elliptically polarized, partially elliptically polarized and partially plane polarized light give with Iceland spar two streams of, in general, unequal intensity, neither of which can be made to vanish. They may be differentiated by first passing the light through a quarter-wave plate with its axis parallel or perpendicular to the plane of maximum polarization: for elliptically polarized light thereby becomes plane polarized and one of the streams is extinguished on rotating the rhomb; but with the other two kinds of light this is not the case, and the light is partially plane or partially elliptically polarized according as the plane of maximum polarization remains the same or is changed.
Colours of Crystalline Plates.—It was known to E. T. Malus that the interposition of a doubly retracting plate between a polarize and an analyser regulated for extinction has the effect of partially restoring the light, and he used this property to discover double refraction in cases in which the separation of the two refracted streams was too slight to be directly detected. D F. J. Arago in 1811 found that in the case of white light and with moderately thin plates the transmitted light is no longer white but coloured, a variation of brightness but not of tint being produced when the polarize and analyser being crossed are rotated together, while the rotation of the analyser alone produces a change of colour, which passes through white into the complementary tint. This phenomenon was subjected to a detailed investigation by Jean Baptiste Biot during the years 1812 to 1814, and from the results of his experiments Thomas Young, with his brilliant acumen, was led to infer that the colours were to be attributed to interference between the ordinary and extraordinary streams in the plate of crystal. This explanation is incomplete, as it leaves out of account the action of the polarize and analyser, and it was with the purpose of removing this defect that Fresnel and Arago undertook the investigations mentioned above and thus supplied what was wanting in Young's explanation. In Biot's earlier experiments the beam of light employed was nearly parallel: the phenomena of rings and brushes that are seen with] a conical pencil of light were discovered by Sir David Brewster in the case of uniaxal crystals in 1813 and in that of biaxal crystals in 1815.
Let α, β, ψ be the angles that the primitive and final planes of polarization and the plane of polarization of the quicker wave within the plate make with a fixed plane, and let ρ be the relative retardation of phase of the two streams on emergence from the plate for light of period τ. On entry into the crystal the original polarized stream is resolved into components represented by
a cos(ψ−α) cos T, a sin (ψ−α) cos T, T=2πt/τ
and on emergence we may take as the expression of the waves
a cos (ψ−α) cos T, a sin (ψ−α) cos (T−ρ).
Finally after traversing the analyser the sum of the two resolved components is
a cos (ψ−α) cos (ψ−β) cos T+a sin (ψ−α) sin (ψ−β) cos (T−ρ),
of which the intensity is
{a cos (ψ−α) cos (ψ−β) +a sin (ψ−α) sin (ψ−β) cos ρ}2+ a2sin2(ψ−α) sin2(ψ−β) sin2 ρ=
a2cos2(β−α) −a2sin 2(ψ−α) sin 2(ψβ) sin2 12ρ.
When the primitive light is white, this expression must be summed for the different monochromatic constituents. In strictness the angle up is dependent upon the frequency, but if the dispersion be weak relatively to the double refraction, the product sin 2(ψ−α)sin 2(ψ−β) has sensibly the same value for all terms of the summation, and we may write
l=cos2(β−α)Σa2 −sin 2(ψ−α) sin 2(ψ−β)Σa2 sin2 12ρ.
This formula contains the whole theory of the colours of crystalline plates in polarized light. Since the first term represents a stream of white light, the plate will appear uncoloured whenever the plane of polarization of either stream transmitted by it coincides with either the primitive or final plane of polarization. In intermediate cases the field is coloured, and the tint changes to its complementary as the plate passes through one of these eight positions, since the second term in the above expression then changes sign If, however, the primitive and final planes of polarization be parallel or crossed, the field exhibits only one colour during a complete revolution of the plate. The crystalline plate shows no colour when it is very thin, and also when its thickness exceeds a moderate amount. In the former case the retardation of phase varies so little with the period that the intensity is nearly the same for all colours; in the latter case it alters so rapidly that for a small change in the period the intensity passes from a maximum or a minimum, and consequently so many constituents of the light are weakened and these are so close to one another in frequency, that the light presents to the eye the appearance of being white. The true character of the light in this case may be revealed by analysing it with a spectroscope, when a spectrum is obtained traversed by dark bands corresponding to the constituents that are weakened or annulled. The phenomenon of colour may, however, be obtained with thick plates by superposing two of them in a suitable manner, the combination acting as a thicker or a thinner plate according as the planes of polarization of the quicker waves within them are parallel or crossed. In this way at delicate test for slight traces of double refraction is obtained. When the retardation of phase for light of mean period is π or a small multiple of π a crystalline plate placed between a crossed polarizer and analyser
