Fig. 3: The spatio-spectral and spatiotemporal structure of generated microwave SNHPs.

Spatial frequency spectra at z = 0.4 m of a1 canonical, a2 simulated, and a3 measured transverse electric component E_y of the SNHPs with parameters q₁ = 0.03 m, q₂ = 20q₁, and \(\ell=1\). ρ and f denote the radius and frequency, respectively. The color indicates the normalized intensity of the spectrum. Sim simulation, Exp experiment, Exp-fit fitted curve of the experimental data. The maximum positions of each frequency point corresponding to canonical, simulated, and measured SNHPs are highlighted in (a3). SNHPs exhibit a wide bandwidth. As the frequency increases, the spatial spectrum becomes narrower and its peak shifts toward the central axis ρ = 0. Spatiotemporal field distributions of the transverse component E_y and longitudinal component E_z for the canonical, simulated, and measured SNHPs are shown in (b1–b3, c1–c3), respectively. The blue and red lobes represent positive and negative electric field components, respectively. Both simulated and measured E_y components exhibit a double-lobe single-cycle helical topology, and E_z components display a four-lobe helical topology, akin to the canonical SNHPs. The concurrence (con) and entanglement of formation (EoF) evolution of the measured transverse electric field components d1 E_x and d2 E_y after radiating from the spiral emitter suggests that the generated SNHPs evolve into states with enhanced space-time nonseparability during propagation. The state-tomography matrix of E_x and E_y of the generated non-transverse SNHPs are shown in (d1, d2), respectively. In (d1, d2), \(|{\lambda }_{1}\rangle\)–\(|{\lambda }_{6}\rangle\) and \(|{r}_{1}\rangle\)–\(|{r}_{6}\rangle\) represent the spectral states and spatial states, respectively. The color indicates the normalized intensity of the state-tomography matrix. The measured fidelities (F) exceed 0.8, confirming the consistency with canonical non-transverse SNHPs.