The Poincaré sphere representation, a geometrical method for solving problems involving the propagation of polarized light through birefringent and optically active media, is applied to several electrooptic liquid crystal problems. The emphasis is on the twisted nematic case, for which the quiescent state solution was given by Mauguin in 1911. The Poincaré construction shows that the normal modes for the undeformed twisted nematic layer are slightly ellipticity. polarized and suggests convenient experiments for measuring the ellipticity. For the field-activated state, a construction is indicated as an alternative to matrix-multiplication methods.
J. E. Bigelow and R. A. Kashnow, "Poincaré sphere analysis of liquid crystal optics," Appl. Opt. 16, 2090-2096 (1977)