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 $\cos^2\Delta\theta$ .

Figure 1. An optical bench (beam through each element with the polarization ellipse at every stage), the input-versus-output ellipse, and the stage-by-stage path on the Poincare sphere. Method: 2x2 complex Jones matrices; Stokes vector and ellipse parameters at each stage.

input

input angle0

element 1

elem 1 axis45

element 2

WHAT TO TRY

Send linear light through a quarter-wave plate at 45 degrees: the field ellipse opens from a line into a circle, and the Poincare point swings from the equator up to a pole.
Chain a second element: each wave plate or rotator slides the state along a great circle of the Poincare sphere, the geometric picture of how Jones matrices compose.
Read the Ex(t) and Ey(t) traces: their phase difference is the polarisation, zero for linear, a quarter cycle for circular, anything between for elliptical.