The constants of an elliptical vibration are determined by transforming it either to a rectilinear vibration or to one of small, known ellipticity and then measuring the azimuth of the transformed vibration. From these measurements the constants of the elliptical vibration can be computed or ascertained by graphical methods. For this transformation, crystal plates of known thickness and birefringence are used. In each one of a series of transformations the incoming vibration is referred to the axes (α′, γ′) of the crystal plate along which the vibrations take place on passage through the plate; the phase lag or lead, introduced by the plate, is then added and the form of the vibration emerging from the plate thus ascertained. This procedure is repeated for each crystal plate and for the analyzer. Poincaré showed in 1892 how the relations of any given elliptical vibration can be represented by a point on a sphere or in a stereographic projection. His graphical method of representation is now in general use. In this connection a specially prepared angle or stereographic projection chart, as a base, greatly facilitates the graphical solutions. A brief description of this chart and of the methods developed for its use, as applied to the derivation of the equations and to the solution of problems in elliptical vibrations, is presented.
Throughout the paper the effect, on a given elliptical vibration represented by a point in the projection, of changing from one set of reference coordinate axes to a second set is obtained in the projection by a rotation about the vertical, N—S, axis through twice the angle between the sets. Similarly the effect of the introduction of a phase angle, 4, by a crystal plate on a given elliptical vibration, referred to the vibration axes (α′, γ′) of the crystal plate and represented by a point in the projection, is obtained by a rotation of the projection about the horizontal, E—W, axis through the angle, ø.
F. E. WRIGHT, "A SPHERICAL PROJECTION CHART FOR USE IN THE STUDY OF ELLIPTICALLY POLARIZED LIGHT," J. Opt. Soc. Am. 20, 529-564 (1930)