Visualization of vortex laser angular momentum distribution. Panels (a) and (b) show the simulated intensity patterns |H| and H_z of a mode inside the vortex laser cavity at Δn_R = Δn_I. The weak fluctuation of the |H| and the periodicity of the H_z clearly show a 16-order traveling mode. (c) Ratio of CW and CCW component |α_m|²/|α_−m |² as a function of real index modulation Δn_R. The imaginary index modulation is fixed at Δn_I = 0.01. The mode shows clear chirality in the vicinity of the exceptional point (Δn_R = Δn_I). The CCW component is about 484 times larger than the CW component at exceptional point, which indicates a nearly perfect traveling wave mode. (d) Angular momentum distribution |α_m|² of the whispering-gallery mode at exceptional point (Δn_R = Δn_I = 0.01). The outer sidewall grating elements (q_OWG = 18) couple the dominated CW mode to the vertically emitted vortex beams with orbital angular momentum index m = −2 (16–18).