The Majorana representation of polarization, and the Berry phase of light

Abstract

Abstract A natural geometric representation of the polarization of light with fixed propagation direction is a dot on a sphere in an abstract space: the Poincaré sphere. If the direction of propagation is also included as a variable, a different description, given here, is natural. It is taken from quantum mechanics (from the Majorana picture of a spin system), spin one in the case of light. It characterizes polarized light by two dots on a unit sphere in the real space of directions (i.e. by two unit vectors). The direction of propagation is their bisector (or its reverse). Projecting the two dots onto the plane perpendicular to this direction gives the two foci of the polarization ellipse (which lies in this plane and has a unit semimajor axis). As an application of this picture the geometric Berry phase for light is calculated. The result accords with the quantum spin-1 formula of Bouchiat and Gibbons, and with the prescription for finding the geometric phase for light given by Bhandari.