Fig. 2

Symmetry-broken hyperbolic polaritons configured by an in-plane dipole source. a Theoretical schematics. Panel 1: Calculated angular spectrum \(|{E}_{z}^{p}|\) in momentum space induced by an in-plane dipole p with polarization angle φ. Red line and yellow dashed line represent maximum and minimum contours of \({|E}_{z}^{p}|\), respectively. Panel 2: Calculated \(|{E}_{z}^{p}|\) overlapped with the dispersion of a hyperbolic system. Thick red curves denote the iso-frequency contours. 2θ is the angle between the asymptotes. Gray regions represent the absence of the effect of \({|E}_{z}^{p}|\). Panel 3: Projected hyperbolic dispersion onto \(|{E}_{z}^{p}|\). b Calculated φ- and θ-dependence of \(|{\Delta E}_{z}|\). Black or green dashed lines represent the occurrence conditions of symmetry-broken or symmetric states, respectively. White circles denote where the highest degree of symmetry-breaking is achieved. c–e Simulations of absolute value, real part and Fourier transform of E_z marked by yellow triangle patch. The yellow dashed line represents the \({{|E}_{z}^{p}|}_{\mathrm{min}}\) of the Panel 1 in a. f Simulated electric field intensity |E_z| as a function of φ in calcite (θ = 45° at 1470 cm⁻¹). g Amplitude profiles extracted along the dashed lines in f. h Calculated electric field intensities \(|{E}_{z}^{\mathrm{I}}|\), \(|{E}_{z}^{\mathrm{II}}|\) for peaks I, II and their difference as a function of φ