Visualization and calculation of polarized light I The polarization ellipse, the Poincaré sphere and the hybrid polarization sphere

We describe the historical and mathematical development of the polarization ellipse and the Poincaré sphere. We point out the limitations of the Poincaré sphere in its present use. To overcome these limitations we describe a new polarization sphere that we call the hybrid polarization sphere. This name is used because phase shifting and rotation of polarization components are described by small circles. Furthermore, longitudinal and latitudinal great circles are introduced so that the coordinates of a point on the sphere can be read. The hybrid polarization sphere is described and applied to polarizers, wave plates, and rotators. As a result, the hybrid polarization sphere can be used for both visualization and calculation and enables the difficulties associated with the Poincaré sphere to be overcome.