Poincaré sphere
The Poincaré sphere is a tool for representing the polarization states of electromagnetic waves such as light . Each polarization state corresponds to a point on the sphere , with fully polarized states on the surface, partially polarized states within the sphere and the unpolarized state in the center. Linear polarizations are at the equator of the sphere, circular polarizations at the poles, and elliptical polarizations in between. Orthogonal polarizations are located opposite one another on the spherical surface. The Poincaré sphere is named after the physicist Henri Poincaré .
Mathematical description
Polarization states of electromagnetic waves can be described, among other things, with the aid of standardized Stokes parameters , where the three parameters S 1 , S 2 and S 3 are required for complete definition . These parameters span a three-dimensional vector space in which the polarization states are located as points. The Stokes vectors of all physically possible polarizations are then located within a unit sphere, the Poincaré sphere.
The following polarizations are located at the intersection of the coordinate axes with the spherical surfaces:
- H: linearly horizontally polarized
- V: linearly vertically polarized
- D: linearly diagonally polarized (sometimes also "/" or P for plus )
- A: linearly antidiagonally polarized (sometimes also "\" or M for minus )
- R: right-handed circularly polarized (sometimes also RHCP)
- L: left-handed circularly polarized (sometimes also LHCP)
The pure polarization states correspond to the points on the surface of the sphere , the mixed polarizations correspond to the interior. Inwardly, the purity continuously decreases until the unpolarized state in the center.
application
The Poincaré sphere is suitable for the intuitive illustration of polarization states and, in particular, of transformations between different polarizations. Similar polarization states are always adjacent and continuous transformations, such as when a wave passes through a birefringent medium, correspond to continuous displacements on the sphere. Linear transformations of the polarization correspond to a rotation on the Poincaré sphere around a fixed axis of rotation. Examples:
- A lambda / 2 plate rotates the Poincaré sphere by 180 °. The axis of rotation lies on the equator and is aligned with the orientation of the wave plate.
- A lambda / 4 plate rotates the Poincaré sphere by 90 °. The axis of rotation lies on the equator and is aligned with the orientation of the wave plate.
- An electro-optical modulator rotates the Poincaré sphere through a variable angle. The axis of rotation lies on the equator and is aligned with the orientation of the electro-optical modulator.
Relationship to other representations
The Poincaré sphere is mathematically equivalent to the Bloch sphere , which illustrates the Hilbert space of the possible states of a quantum mechanical two-state system ( qubit ). The orientation with which the two spheres can be mapped to one another depends on which two orthogonal (opposite) polarizations are selected as the base states. This mapping corresponds to using the polarization of a photon as a qubit.
If the polarization is described mathematically equivalent with Jones vectors instead of Stokes parameters , the Jones vector corresponds precisely to the Bloch vector, provided that the polarizations H and V are selected as the base states on the Poincaré sphere.
If you transfer points of the Poincaré sphere to the Riemann number sphere , you get a complex number that describes the amplitude and phase relationship of horizontal and vertical oscillation.
Individual evidence
- ^ Dieter Meschede: Optics, light and laser. Wiesbaden 2005, ISBN 978-3-8351-0143-2 , p. 62.
- ↑ Georg A. Reider: Photonics: An introduction to the basics. Vienna 2012, ISBN 978-3-7091-1520-6 , p. 32.