Stability of the vortex-beam output in the lasing process under pumping. (a) The background gain dependence of the cavity quality factor for a vortex beam laser at exceptional point (Δn_R = Δn_I = 0.01) with N_g = 19. The uniform pumping gain of the InGaAsP ring is mimicked by increasing the imaginary part of background refractive index n_I. The quality factor is about 365 for the cavity without gain. With the increasing of the gain coefficiency, the cavity quality factor increases by orders of magnitude, indicating that the loss is compensated by the gain. (b) Ratio of the CW and CCW components |α_m |²/|α_−m|² as a function of background refractive index n_I. The black dots show the main component of the mode is CW mode, which is almost unchanged with the increase of the background refractive index n_I. The CW traveling mode is coupled to a vortex beam with azimuthal quantum number m = −2 (see the red dots).