Wigner small-d matrix ($d_{m^{\prime }m}^j(\theta )=⟨j,m^{\prime }|e^{-i{J}_y\theta }|j,m⟩$)

The rotation matrix about the $y$ axis in a spin-$j$ angular-momentum representation. For $j=1$, $d_{-1,+1}^1=(1-\cos \theta )/2$; its square gives the polar dependence of a transition between those spin projections.