Vector spherical harmonics (Multipoles). Rotations. Total angular momentum. Helicity.

Literature:

1. Bohren, C. F.; Huffmann, D. R. (2010). Absorption and scattering of light by small particles. New York: Wiley
2. Stratton, Julius Adams. Electromagnetic Theory. 2015
3. Khersonskii, V. K., et al. Quantum Theory Of Angular Momentum. World Scientific Publishing Company, 1988, doi:10.1142/0270.
4. Tung, Wu-Ki. Group Theory in Physics. 1985
5. https://en.wikipedia.org/wiki/Vector_...
6. Gladyshev, Sergey, et al. Phys. Rev. B, vol. 102, no. 7, 3 Aug. 2020, p. 075103, doi:10.1103/PhysRevB.102.075103 (picture is taken from here)
7. Vavilin, Maxim and Ivan Fernandez-Corbaton. "The polychromatic T-matrix.” J. Quant. Spectrosc. Radiat. Transfer, vol. 314, 1 Feb. 2024, p. 108853
8. Toftul, Ivan, et al. "Nonlinearity-Induced Optical Torque." Phys. Rev. Lett., vol. 130, no. 24, 15 June 2023, p. 243802 (see Supplementary)
9. Daniel Zwanziger. Quantum Field Theory of Particles with Both Electric
and Magnetic Charges. Phys. Rev., 176:1489{1495, Dec 1968.
10. Hergert, Wolfram and R. Matthias Geilhufe. Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica. Wiley-VCH, 2018, (Rotations of vector functions)
11. Gladyshev, Sergey, et al. "Symmetry analysis and multipole classification of eigenmodes in electromagnetic resonators for engineering their optical properties." Phys. Rev. B, vol. 102, no. 7, 3 Aug. 2020, p. 075103, https://arxiv.org/pdf/2002.11411 Multipolar decomposition of eigenmodes
12. Nikitina, Anastasia and Kristina Frizyuk. "Achiral Nanostructures: Perturbative Harmonic Generation and Dichroism Under Vortex and Vector Beams Illumination." Adv. Opt. Mater., vol. 12, no. 25, 1 Sept. 2024, p. 2400732, https://doi.org/10.1002/adom.202400732



00:00 Intro
00:22 Expressions for vector spherical harmonics
01:42 Far-field
02:27 Illustration. How do the multipoles look like.
03:05 Behavior of spherical functions and VSH under rotations
04:14 Complex multipoles
04:50 Total angular momentum
05:12 projection of TAM
06:06 Helicity