Fig. 1: Topological vortex multiplication via sublinear SPhP dispersion.

a, Dispersion-driven tuning for free-space and polaritonic OVs based on azimuthally varying phase delay. The strong sublinear dispersion of surface polaritons results in a smaller frequency increase Δω_SP for the same Δk compared with the linear case, Δω. b, Changing the excitation frequency so that the momentum is increased by an integer number nk leads to OAM multiplication. The three frequencies are chosen such that the polariton momentum is k₁, 2k₁ and 3k₁. σ = ±1 is the spin state of the incident beam. c, Sketch of the transmission sSNOM setup used to probe the near field of polaritonic OVs. d, Calculated dispersion of the anti-symmetric mode of a 100-nm-thick SiC membrane. The insets show the simulated near-field phase maps for an SPhP vortex at three different frequencies launched by the same VG designed at ω = 880 cm⁻¹ with L = 1 and excited by LCP light. Scale bar, 2 μm. e–g, Overlap integral at ω = 924 cm⁻¹, ω = 910 cm⁻¹ and ω = 880 cm⁻¹ used to evaluate the vortex purity. The insets show the zoomed-in views of the maps shown in d. Scale bar, 1 μm. h–k, Overlap integral for orders ℓ = 4, ℓ = 3 and ℓ = 2 as a function of excitation frequency. λ_d indicates the SPhP wavelength used for VG design.