Poincaré sphere representation of the fixed-polarizer rotating-retarder optical system

The trajectory of the polarization state of a monochromatic light beam after it passes through a fixed linear polarizer and a rotating linear retarder of arbitrary retardance delta is determined on the Poincaré sphere. The three-dimensional figure-8 contour is shown to be the line of intersection of a right-circular cylinder with the sphere. The cylinder is parallel to the polar (S3) axis, touches the sphere at the equator (at the point that represents the linear polarization transmitted by the fixed polarizer), and has a radius r = sin2(delta/2). Projections of the trajectory in the coordinate planes of the normalized Stokes parameter space (s1, s2, S3) are also determined.