Spin-weighted spherical harmonics are generalizations of standard spherical harmonics and like them are complex functions on the sphere. The multipole expansions of an electromagnetic field can be expressed in terms of spin-weighted spherical harmonics. They are particularly important in the theory of gravitational waves.

For spin weight , the spin-weighted spherical harmonics become identical to the spherical harmonics. The case of spin weight is important for describing gravitational waves. In this Demonstration you can choose different values of the spin weight to see the angular distribution in space for different and modes.

J. J. G. Scanio, "Spin-Weighted Spherical Harmonics and Electromagnetic Multipole Expansions," Am. J. Phys., 45(2), 1977 pp. 173–178.

3) Advanced level reference:

J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, "Spin- Spherical Harmonics and ," J. Math. Phys., 8(11), 1967 pp. 2155–2161.