Quantum Optics for Engineers

Appendix H: Poincaré’s Space

H.1  Poincaré’s Space

A useful tool in polarization notation is derived from Poincaré’s sphere (Poincaré, 1892).

This sphere, depicted in Figure H.1, has three axes 1, 2, and 3. Axis 2 is analogous to the usual Cartesian axis x, axis 3 is analogous to the usual Cartesian axis y, and axis 1 is analogous to the usual Cartesian axis z, that is,

$1\to z$

$2\to x$

$3\to y$

Adopting the notation of Robson (1974), the radius of the sphere is denoted by I. The angular displacement in planes 1–2 is 2ψ and the angular displacement between planes 1–2 and axis 3 is denoted by 2χ. In this system, the points P₁, P₂, P₃ are given by

$P_1=I\text{cos}2\chi \text{cos}2\text{ψ}$

(H.1)

$P_2=I\text{cos}2\chi \text{sin}2\text{ψ}$

(H.2)

$P_3=I\text{sin}2\chi$

(H.3)

These are known as the ...