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

These are known as the Stokes parameters. In addition to Poincaré’s sphere, the polarization space described here is also known as Bloch’s sphere (Pelliccia et al., 2003).

FIGURE H.1
Poincaré’s sphere.

References

Pelliccia, D., Schettini, V., Sciarrino, F., Sias, C., and De Martini, F. (2003). Contextual realization of the universal quantum cloning machine and of the universal-NOT gate by quantum-injected optical parametric amplification. Phys. Rev. A 68, 042306.

Poincaré, H. (1892). Théorie Mathématique de la Lumière, Vol. 2. Corré, Paris, France.

Robson, B. A. (1974). The Theory of Polarization Phenomena. Clarendon Press, Oxford, U.K.