Fig. 3: Chiral symmetry breaking.

a, A simple model of asymmetric pumping for lifting the degeneracy of CW and CCW modes. Two circular pump beams are applied with a relative angle θ, while keeping the same distance to the centre of the cavity fixed as 2.5 μm and the radii fixed as 1 μm and 2 μm, respectively. \(\tilde{n}\) and \({\tilde{n}}_{1,2}\) are the complex indices of the PhC cavity area, which are unpumped and pumped by two beams, respectively. b, The evolution of complex eigenfrequencies in parameter space, showing the emergence of an EP. The parameters are set as \(\tilde{n}=3.25-0.01i\), \({\tilde{n}}_{2}=3.25+0.01i\), \(Re({\tilde{n}}_{1})=3.25\) and θ = 100.15°; the EP is found at \(Im({\tilde{n}}_{1})=0.035\), at which the two eigenstates collapse to single CW chirality. The chirality degrades when the parameters deviate from the EP in the region where the real eigenfrequencies are degenerate (orange-shaded), in which the difference in the imaginary parts offers mode selection for single-mode lasing. Imag.(ñ₁) represents the imaginary part of ñ₁. c, The electrical field strength ∣E∣ of the eigenstate at the EP, exhibiting a donut shape in real space. d, The evolution of snapshot magnetic field H_z in a time interval of 1/4T, clearly showing the rotation motion along CW direction. All results are calculated from numerical simulations (COMSOL Multiphysics).

Source data