Fig. 2: Exceptional point in a gain-loss coupled cavity.

a A coupled 1D cavity separated by a distributed Bragg reflector (DBR), with gain in the left cavity and absorption in the right cavity. Gray-scale colors indicate the cold-cavity permittivity profile ε_c(x). Orange and blue shades show the gain and absorption profiles D_p(x) and σ(x), respectively. b, c The two relevant eigenvalues, \({\tilde{\omega }}_{0}\) and \({\tilde{\omega }}_{1}\), of the linear operator \(\hat{O}\left(\omega \right)\) in Eq. (4) with a linear gain D₀(x) = D_p(x), as a function of the pumping strength \({D}_{\max }\) and the length of the passive cavity, L₂ = 1340 nm + Δ. The two eigenvalues meet at an EP (green circle). The absorption in the passive cavity is σ/ε₀ = 4.9 ps⁻¹. The red and blue curves indicate \({\tilde{\omega }}_{0}\) and \({\tilde{\omega }}_{1}\) with Δ = 0. d Eigenvalue trajectories on the complex-frequency plane for Δ = 0, with the orange arrows indicating the directions of increasing \({D}_{\max }\). Open circles indicate \({D}_{\max }=0\), and filled red and blue circles indicate the first lasing threshold \({D}_{\max }={D}_{1}^{{{\rm{th}}}}\) where \({\tilde{\omega }}_{0}\) reaches the real-frequency axis. Dashed lines show the would-be above-threshold trajectories in the absence of gain saturation.