From Poincaré sphere (Bloch sphere) to Projective space

Описание: List of references

K. Frizyuk, E. Menshikov, and M. Spera, A novel geometric phase for optical beams, https://arxiv.org/abs/2602.05427

Takenaka, Hiroshi. "A unified formalism for polarization optics by using group theory." Nouv. Rev. Opt., vol. 4, no. 1, 1 Jan. 1973, p. 37, doi:10.1088/0335-7368/4/1/304.

H.K. Urbantke, The Hopf fibration—seven times in physics, Journal of Geometry and Physics 46 (2003) 125–150.

https://arxiv.org/abs/1303.4496 -- del Castillo, G. F. Torres and I. Rubalcava-Garcia. "The Jones vector as a spinor and its representation on the Poincar´e sphere."

https://arxiv.org/abs/2107.10810 -- Sugic, Danica, et al. "Particle-like topologies in light." Nat. Commun., vol. 12, no. 6785, 22 Nov. 2021, pp. 1-10, doi:10.1038/s41467-021-26171-5.

https://arxiv.org/abs/1508.00920 Oblak, Blagoje. "From the Lorentz Group to the Celestial Sphere."

Jones, G. A. and D. Singerman. Complex Functions. Cambridge UP, 1987.

https://arxiv.org/abs/2303.17112 Saito, Shinichi. "Spin of Photons: Nature of Polarisation." arXiv, 30 Mar. 2023

(btw, it is better to use helicity or polarization for light, but not spin. Spin is not well defined for photons)

00:00:00 Intro. Poincare sphere
00:00:50 Each point --- polarization state of a plane wave
00:01:08 Why does it look like this
00:01:30 Spoiler!
00:01:45 Def. Equivalence relation
00:02:13 Def. Equivalence class
00:02:26 Def. Quotient set
00:02:50 Def. Real projective space
00:03:40 ℝℙ¹ (real projective line) Example
00:04:37 Stereographic projection
00:05:40 Be careful
00:05:50 ℝℙ² (real projective plane) Example
00:06:09 Boy's surface (immersion)
00:06:40 Complex projective space. (Complex projective line)
00:07:20 Stereographic projection in 2D
00:07:48 Finally optics!!!
00:08:47 Projecting polarization state to a sphere somehow
00:09:37 Is Riemann sphere indeed the same sphere? Proof
00:10:36 Stokes parameters map 2d complex space (and 3-sphere) to the 2-sphere