Fig. 1: Concept of lasing EP.

For reference, a linear PT symmetric coupled-cavity system is presented in (a). The case of coupled identical laser cavities with an ideal gain medium is plotted in (b): here, nonlinear saturation effects are taken into account but the population inversion does not induce any refractive index change, therefore there is no pump-induced frequency shift of the resonant modes (α = 0 in Eqs. (1)). Note that the EP becomes a nonlinear (pitchfork) bifurcation point (marked by the label BP). c A nonzero phase-amplitude coupling in the semiconductor cavity (nonzero α-factor) induces an asymmetric blueshift, with a net detuning Δω between the two cavities and thus breaks their parity symmetry. As a result, it impedes the formation of the EP. The effective PT symmetry (g₁ = −g₂, g_j = −κ + (n_j − n₀)βγ_∥/2, j = 1, 2), and hence the formation of an EP can be restored if the (active) blueshift is compensating for by using a static redshift (i.e., introduced in the design from the beginning) δω as shown in (d). The colors of the disks represent the sum of the frequency shift due to the carriers, Δω_1,2 = α(n_1,2 − n₀)βγ_∥/2, and the fabrication. Here we take P_tot = 3P₀, where P₀ is the threshold of a single cavity. In the bifurcation diagrams, H represents the Hopf bifurcation, LP is a limit point or fold, where two steady-state solutions annihilate each other (note that the slope of the curve at these points is infinite), and BP refers to a branch point bifurcation.