Fig. 3: Coupling in the twisted bilayer system.
From: Chiral orbital lasing in a twisted bilayer metasurface
a Coupling scenario of four unperturbed single-layer modes \(\left|{\psi }_{CW,\,CCW}^{\,u,\,l}\right\rangle\), illustrating three distinct coupling paths as intra-layer cross-coupling (\({\kappa }_{1,2}^{\,{{{\rm{intra}}}}}\), orange arrows), inter-layer cross-coupling (\({\kappa }_{1,2}^{\,{{{\rm{inter}}}}}\), blue arrows), and inter-layer direct-coupling (\({\kappa }_{\,{{{\rm{direct}}}}}^{{{{\rm{inter}}}}}\), purple arrow). The coupling coefficients are complex due to the non-Hermiticity. b Complex bands of hybridized modes \(\left|{\psi }_{1-4}\right\rangle\) are tuned by varying the intra-layer cross-coupling strength \({\kappa }_{1}^{\,{{{\rm{intra}}}}}\). An exceptional point (EP) emerges at \({\kappa }_{1}^{\,{{{\rm{inter}}}}}=-{\kappa }_{1}^{{{{\rm{intra}}}}}\), where the eigenstates collapse. At the EP, the system reaches the highest degree of chirality, showing CW rotation existing in both layers. c Two representative chiral modes are generated through helical and non-Hermitian couplings with eigenvectors V = [0, 1, 0, −1]T and V = [0, 1, 0, 1]T, respectively. Here, the upper layer is fixed to CW rotating and their z-phases are 0 (red arrow) or π (blue arrow). The mode V = [0, 1, 0, 1]T corresponds to the configuration shown in (b).
