Spin-weighted spherical harmonic (${}_s{Y}_{\ell m}$)

The angular basis ${}_s{Y}_{\ell m}$ for a spin-weight-$s$ field generalizes ordinary spherical harmonics. It has $\ell \ge |s|$. For scalar-source CMB polarization about a wavevector, only $m=0$ survives and ${}_{\pm 2}{Y}_{20}\propto {\sin }^2\theta$.

On a spin-$s$ field $f$, the spin-raising and lowering operations are $ðf=-{\sin }^s\theta ({\partial }_{\theta }+i\csc \theta {\partial }_{\phi })({\sin }^{-s}\theta f)$ and $\overline{ð}f=-{\sin }^{-s}\theta ({\partial }_{\theta }-i\csc \theta {\partial }_{\phi })({\sin }^s\theta f)$. The intermediate spin changes after each application. For an axisymmetric scalar $f(\mu )$, either operation applied twice yields $(1-{\mu }^2)f^{\prime \prime }(\mu )$.