Exceptional points in the vortex laser cavity. (a) A general form of refractive-index modulation in a micro-ring cavity. The index modulation consists of two sets of periodic scattering elements, which are essential to realize exceptional points in the view of interference. Panels (b) and (c) show the relations the parameters need to satisfy simultaneously to achieve exceptional points on the condition of ϕ₁ = 0 and ϕ₂ = π/2. Panels (d) and (e) are the real part and imaginary part of the eigen-values in the case of one grating with pure real part n_R (n_R = Δ₁ e ^iϕ₁, ϕ₁ = 0) while the other grating with pure imaginary n_I (n_I = Δ₂ e ^iϕ₂, ϕ₂ = π/2) index modulation and keeping 2mφ₀ = π/2. This is the case of PT-symmetrical refractive index modulation. In the calculation, azimuthal order m = 16, refractive index of the cavity n₀ = 2.67, Δn_R = nπ/16 ≤ φ ≤ nπ/16+π/32, Δn_I = nπ/16+π/64 ≤ φ ≤ nπ/16+3π/64, n = 0,1,2,…,31. The inner radius and the width of the vortex laser cavity is 1500 nm and 500 nm respectively.