= Helicity decomposition of a transverse Fourier mode
{title2=$m^{(\pm)}=(\hat x\pm i\hat y)/\sqrt2$}
For $\mathbf k=k\hat z$, these complex vectors form an orthonormal basis of the transverse plane using the Hermitian <inner product>. They have opposite phases under rotations about $\hat z$, so they describe the two <helicity> components. Since $\mathbf e\cdot m^{(\pm)}=\sin\theta e^{\pm i\varphi}/\sqrt2$, transverse-vector sources carry only the angular sectors $m=\pm1$ of <spherical harmonics>. Streaming by $ik\cos\theta$ preserves $m$ and couples neighboring angular degrees.
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