Forward cone and regular north-patch helicity gauge

Assume the Axiom of Choice and fix constants $0<{\kappa }_{-}<{\kappa }_{+}<\mathrm{\infty }$ and $0<{\theta }_{\max }<\pi \mathrm{/}2$. Let $M={\mathbb{R}}^3\setminus \{0\}$, $r=\mathrm{\mid }\mathbf{k}\mathrm{\mid }$, $\mathbf{n}=\mathbf{k}\mathrm{/}r$, and use the positive $z$ axis as the propagation axis. Set $\theta (\mathbf{k})=\arccos (n_z)\in [0,\pi ]$ and let $\phi \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ be the azimuth on $0<\theta <\pi$. Use the transverse Hilbert space ${\mathcal{H}}_T^{\nu }=\{f\in L^2(M,d\nu ;{\mathbb{C}}^3):\mathbf{k}\cdot f(\mathbf{k})=0\nu \text{-almost everywhere}\}$ with $d\nu =d^3\mathbf{k}\mathrm{/}(2\mathrm{\mid }\mathbf{k}\mathrm{\mid })$ and the helicity projectors $P_{\sigma }(\mathbf{n})=(P(\mathbf{n})+\sigma h(\mathbf{n}))\mathrm{/}2$, where $P=I-\mathbf{n}{\mathbf{n}}^T$, $h=i(\mathbf{n}\times \cdot )$, and $\sigma \in \{+1,-1\}$. A normalized helicity-$\sigma$ packet in the forward cone is a vector $f\in {\mathcal{H}}_T^{\nu }$ such that $\mathrm{\parallel }f{\mathrm{\parallel }}_{\nu }=1$, $P_{\sigma }(\mathbf{n}(\mathbf{k}))f(\mathbf{k})=f(\mathbf{k})$ almost everywhere, and $f(\mathbf{k})=0$ almost everywhere unless ${\kappa }_{-}\le \mathrm{\mid }\mathbf{k}\mathrm{\mid }\le {\kappa }_{+}$ and $0\le \theta (\mathbf{k})\le {\theta }_{\max }$. The paraxial regime in this aperture-parameterized family is the small-angle limit ${\theta }_{\max }\downarrow 0$; at a fixed aperture below $\pi \mathrm{/}2$ this definition gives a forward cone, and finite-aperture estimates retain their ${\theta }_{\max }$ dependence.

On the north patch $U_N=S^2\setminus \{-e_z\}$, write ${\mathbf{e}}_{\theta }=(\cos \theta \cos \phi ,\cos \theta \sin \phi ,-\sin \theta )$ and ${\mathbf{e}}_{\phi }=(-\sin \phi ,\cos \phi ,0)$. Define $e_{\sigma }=({\mathbf{e}}_{\theta }+i\sigma {\mathbf{e}}_{\phi })\mathrm{/}\sqrt{2}$ and $e_{\sigma }^N=e^{i\sigma \phi }{e}_{\sigma }$, with its smooth extension at $+e_z$. The south gauge is $e_{\sigma }^S=e^{-i\sigma \phi }{e}_{\sigma }$; the transition is $e_{\sigma }^N=e^{2i\sigma \phi }{e}_{\sigma }^S$, and neither helicity line has a global smooth unit frame. Relative to $e_{\sigma }^N$, a scalar coefficient with winding $\mathrm{\ell }\in \mathbb{Z}$ has the form $c(r,\theta ,\phi )=g(r,\theta )e^{i\mathrm{\ell }\phi }$, where $g$ is independent of $\phi$. If its support reaches the axis, the sufficient regular form is $c=b(r,n_z)(\sin \theta {)}^{\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }}{e}^{i\mathrm{\ell }\phi }$ near $\theta =0$, where $b\in C^{\mathrm{\infty }}(V)$ for an open $V\subseteq {\mathbb{R}}^2$ containing the relevant $(r,n_z)$ values. If the support avoids the axis, any smooth envelope there makes the coefficient smooth for every integer $\mathrm{\ell }$. The winding $\mathrm{\ell }$ is a label in this chosen patch, not a global orbital label. Use the rotation convention $U(R_z(t))f(\mathbf{k})=R_z(t)f(R_z(-t)\mathbf{k})=e^{-it{J}_z\mathrm{/}\mathrm{\hslash }}f(\mathbf{k})$.

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