Jones Calculus: Polarization Through Elements

A fully polarized beam is a Jones vector $(E_x,E_y)$ of complex amplitudes; every polarizer or wave plate is a $2\times2$ Jones matrix, and a stack of them is just the matrix product. The left panel traces the polarization ellipse the field draws in a plane; the right is the Poincare sphere, where every state is a point and each element rotates or projects it. A quarter-wave plate at $45^\circ$ to linear light turns it circular (a quarter turn on the sphere); a half-wave plate reflects linear polarization about its axis; a polarizer projects onto a diameter and dims the beam by Malus's law $\cos^2\Delta\theta$.