Poincare sphere - two-dimensional sphere , in Cartesian coordinates determined by the Stokes parameters . In polarizing optics introduced by Henri Poincare in 1892 [1] . In other sections of the physics of this model , the Bloch sphere corresponds. From the homology three-dimensional sphere ( homology sphere ) in physics there remains only the base of the Hopf bundle - the Riemann sphere . Information about the third dimension ( oscillation phase ) is discarded. This projective simplification made it possible to fabricate a model of separation of the phase space of polarizations in the form of a ball, which made it possible to visually calculate specific wave processes. [2]
In mechanics, the Poincare sphere describes the state of small oscillations of a spherical pendulum, a Lissajous figure of the same frequency. [3]
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To each point of the sphere we associate a small oriented circle lying on the sphere centered at this point. The parallel projection of such a sphere onto a plane will translate the circles into all kinds of polarization ellipses. However, each such ellipse occurs twice (which corresponds to the same oscillations of the tension vector, but in antiphase). The Poincare sphere can be obtained by gluing pairs of points of the main meridian located on the same parallel.
Bonding points corresponding to the same polarization. Only the upper hemisphere corresponding to the left polarizations is shown. The azimuthal angle is doubled. The climb tangent also doubles. [four]
The representation of polarized light using a single complex number is obtained by stereographic projection of the Poincare sphere onto the complex plane. [5]
See also
- Wave polarization
- Riemann Orb
- Hopf bundle
Notes
- ↑ Poincare H. Theorie Mathematique de la lumiere, vol. 2, Gauthiers-Villars, Paris, 1892 , ch. 12.
- ↑ HG Jerrapd. Transmission of Light through Birefringent and Optically Active Media: the Poincare Sphere (Eng.) // JOSA : journal. - 1954. - Vol. 44 , no. 8 . - P. 634-640 .
- ↑ V.I. Arnold. Mathematical methods of classical mechanics . - ed. 3. - M. , 1988. - S. 472. (unavailable link) Ch. 2, par. 5, G. Example 1. Small oscillations of a spherical pendulum, D. Example 2. Lissajous figures. pg. 23-25.
- ↑ Sherkliff W. Polarized Light. - M.: Mir, 1965. - S. 264. Ch. 2. Modern methods for describing polarized light, Fig. on page 28.
- ↑ Azzam R., Bashara N. (Azzam, Bashara). Ellipsometry and polarized light. - M.: Mir, 1981. - S. 584. Paragraph. 1.8. Representation of polarized light by points on the Poincare sphere, Fig. 1.22. on page 66.