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 *P*_{1}, *P*_{2}, *P*_{3} are given by

${P}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}I\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}2\chi \text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}2\text{\psi}$ |
(H.1) |

${P}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}I\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}2\chi \text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}2\text{\psi}$ |
(H.2) |

${P}_{3}=I\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}2\chi $ |
(H.3) |

These are known as the ...

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