ABSTRACT
The Jones Pupil and Local Coordinate Systems analyzes the difficult issues of converting from ray trace results defined on spherical surfaces in three dimensions into flat surface representations as a Jones pupil, the same issues involved in printing maps of the earth. The description of the polarization with three-dimensional global coordinates is robust, straightforward to calculate, and provides an excellent method for a computer ray trace. The results however can be difficult to visualize in three-dimension because optical designers view ray tracing results on computer screens and print the information onto paper. Visualizing the spherical waves on a plane surface requires transforming a 3-dimensional vector field onto two dimensions. The most common method to represent the polarization aberrations is as a Jones matrix function in two-dimensional pupil coordinates, the Jones pupil. In order to use Jones pupils properly, the subtleties of local coordinate systems need to be well understood. In this chapter, optimal methods for conversions are presented.
Two principal local coordinate systems are developed: dipole coordinates and double pole coordinates. Dipole coordinates are latitude and longitude vectors on a sphere and can describe the fields radiated by dipoles aligned with lines of longitude.
For high numerical aperture wavefronts, double pole coordinate systems become more convenient since this coordinate system more closely approximates the natural behavior of lenses; double pole coordinates can describe polarization of the spherical wavefront when a lens brings the collimated linearly polarized wavefront to focus. Double pole coordinates also contain a fascinating doubly degenerate singular point.
